Chapter 21
Externalities in Competitive
Markets
At this point you may have gotten the impression that economists believe markets always and
unambiguously result in efficient outcomes — with total surplus maximized when markets operate
without interference from other institutions.1 If this were the case, there would be no efficiency
role for non-market institutions in society, and their only justification would lie in concerns about
the distribution of surplus — concerns about equity and fairness as these relate to the market
allocation of scarce resources. But, while such issues do play an important role in justifying nonmarket institutions (including government), we will in this and the coming chapters investigate
conditions under which non-market institutions are motivated by efficiency rather than equity
concerns. These conditions include all the possible violations of the assumptions underlying the
first welfare theorem (Chapter 15) – including the presence of market power and of asymmetric
information.
Before we get to asymmetric information and market power, however, we will first take a look
at yet a third set of conditions that lead to dead weight losses in the absence of other institutions
– even when markets are perfectly competitive. These conditions are called externalities, and they
arise whenever decisions of some parties in the market have a direct impact on others in ways that
are not captured by market prices. When a firm’s production process emits pollution into the air,
for instance, this pollution potentially has a direct impact on many. Put differently, the emission
of pollution imposes on society costs that are typically not priced by the market and thus are not
costs taken into account by producers unless some other institution imposes those costs on them.
When I decide to get in the car and enter a congested road, I am similarly contributing to the
overall congestion and thus am delaying others from getting to where they want to go, but I don’t
think about others when I make the decision of whether to get in the car. When I play loud music
on my patio at home, my neighbors get to “enjoy” the music as well. These are all examples of
externalities — of “external costs or benefits” that markets do not internalize because the market
participants do not have to pay for them.
1 This chapter builds once again on a basic understanding of the partial equilibrium model from Chapters 14 and
15. Section 21B.3 also builds on the discussion of exchange economies in Chapter 16 but can be skipped if you have
not yet read Chapter 16.
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Chapter 21. Externalities in Competitive Markets
The Problem of Externalities
The essential feature of an externality is then that either costs or benefits of production or consumption are directly imposed on non-market participants. Since non-market participants are neither
demanders nor suppliers of goods, neither market demand nor market supply curves are affected by
such externality costs or benefits. Thus, a competitive market composed of price-taking consumers
and producers continues to produce in equilibrium where demand intersects supply. However, while
the aggregate marginal willingness to pay curve still allows us to measure the benefits consumers
receive from participating in markets and the supply curve still allows us to measure costs incurred
by producers, there are now non-market participants that also incur benefits or costs. Thus, we
can no longer simply use consumer and producer surplus to measure the net-gains for society from
the existence of markets. Put differently, we have to include the externality costs and benefits that
a competitive market ignores in our calculation of overall surplus.
Before we get started, I should note that we will treat consumers and producers as strictly separate in their roles as consumers and producers from their roles as individuals who may incur some
damage or benefit from an externality. We generally lose nothing by making this assumption. Even
if, for instance, a producer whose production causes pollution incurs health problems from pollution, no individual producer will take those costs into account in her production choices because,
in competitive markets, each producer is so small relative to the market that her contribution to
overall pollution is negligible. Thus, we will simply treat all producers as considering only their
own production costs when making decisions, and then lump them in separately with all economic
agents who are hurt by the aggregate level of pollution produced by the industry. In other words,
we will treat producers as individuals who consider their own cost of production when making
supply decisions, and then we will treat the part of that producer that is hurt by the overall level
of pollution as a separate person.
21A.1
Production Externalities
Suppose, then, that we return to the example of an industry that produces “hero cards” but now
we assume that the least-cost production process for producers involves the emission of green-house
gases that contribute to environmental problems. Thus, in addition to the costs of production
that are faced by each of the producers of hero cards, costs of pollution are imposed on others in
society. We will then reconsider how many hero cards would be produced by a social planner who
knows all the relevant costs and benefits and who seeks to maximize social surplus — how much
production would take place if our omniscient and benevolent “Barney” from Chapter 15 would
allocate resources. In our Chapter 15 analysis that excluded production externalities like pollution,
it turned out that “Barney” could do no better than the competitive market. We will now see that
this is no longer true when externalities become part of the analysis.
21A.1.1
“Barney” versus the Market
In Graph 21.1, we begin with the market demand and supply graph for hero cards in panel (a).
Whether there are production externalities or not, the market will then produce xM at price pM ,
with all consumers and all producers doing the best they can in equilibrium. Assuming tastes in
hero cards are quasilinear, consumers then get the shaded blue area in surplus while producers get
the shaded magenta area. If the production of hero cards produces pollution, however, each hero
21A. The Problem of Externalities
761
card that is produced imposes a pollution cost on society, a cost that is borne neither by those who
consume nor those who produce hero cards.
Graph 21.1: Maximizing Social Surplus in the Presence of a Negative Production Externality
Panel (b) of the graph then inserts a green curve labeled “SM C”. This curve represents the
social marginal cost of producing hero cards. It includes the producers’ marginal costs that are
captured in the market supply curve, but it also includes the additional cost of pollution that is
imposed on others. Thus, the social marginal cost curve must lie above the supply curve since it
includes costs in addition to those incurred by producers. It may be that the SM C curve is parallel
to the supply curve — implying a constant marginal cost of pollution for each hero card produced,
or that it diverges from the supply curve — implying that each additional hero card results in a
greater additional pollution cost than the last one. Regardless of how exactly it is related to supply,
however, it is this curve that accurately reflects the society-wide cost of production.
As a result, our omniscient and benevolent “Barney” would then decide to continue to produce so
long as the benefits from production as represented by the marginal willingness to pay of consumers
outweighs the overall cost of additional production for society. Put differently, Barney would
certainly produce the first hero card because there is some consumer to whom this card is worth
more than all the costs incurred by society as measured by SM C, and he would continue to produce
until the green SM C crosses the blue marginal benefit curve. He would not, however, produce any
more than that — because once SM C is higher than the marginal willingness to pay of consumers,
the society-wide cost of additional hero cards is larger than the benefit. Barney then would choose
to produce xB , resulting in an overall surplus for society represented by the shaded green area.
We can already see that the social planner who seeks to maximize overall surplus will therefore
choose less production than will occur in the market. This implies that the market will produce
an inefficiently high level of output in the absence of any non-market institutions that curtail
production. This is clarified even further in panel (c) where we have labeled some areas in the graph
that can now be used to calculate the dead weight loss society incurs under market production.
Area (a + b + c) is equal to the blue consumer surplus (assuming the uncompensated demand is
equal to marginal willingness to pay) in panel (a) while area (d + e + f ) is equal to the magenta
producer surplus from panel (a). Producers and consumers are, in their roles as producers and
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Chapter 21. Externalities in Competitive Markets
consumers, unaffected by the pollution and therefore receive the same surplus as if there was no
pollution. However we also know that, in the presence of pollution, we have to take into account
the overall cost of the pollution that is produced when the market quantity xM is produced. That
area is the difference between the costs incurred by producers and the costs as represented in the
SM C curve — an area equal to (b + c + e + f + g). Thus, we have to subtract that from consumer
and producer surplus to get overall social surplus (a + d − g) under market production. Under
Barney’s benevolent dictatorship, on the other hand, society gets an overall surplus of (a + d) equal
to the green area in panel (b). The market therefore produces a deadweight loss equal to (g).
Exercise 21A.1 Suppose that the “pollution” emitted in the production of hero cards is of a kind that
has no harmful effects for humans but does have the benefit of killing the local mosquito population — i.e.
suppose the pollution is good rather than bad. Would the market produce more or less than Barney?
Exercise 21A.2 Would anything fundamental change in our analysis if we let go of our implicit assumption
that the aggregate demand curve is also equal to the aggregate marginal willingness to pay curve? (Your
answer should be no. Can you explain why?)
21A.1.2
Another Efficient Tax
Our analysis thus far tells us that competitive markets will produce too much in the presence
of negative pollution externalities. As a result, there exists the potential for government policy to
enhance efficiency — and thus reduce or eliminate the deadweight loss from market overproduction.
And we have already seen in Chapter 18 that taxation of goods is one policy tool that can reduce
market output. In the absence of externalities, this is inefficient because the market allocation
of resources was efficient to begin with. Now, however, this reduction of an otherwise inefficient
output level can reduce rather than increase deadweight loss.
Suppose, for instance, you knew both the market demand and supply curves as well as the
optimal production level xB that Barney would choose. This information is depicted in panel (a) of
Graph 21.2. Based on what we learned about taxes and tax incidence in Chapter 18, you can then
easily determine the tax rate t required to reduce market output from xM to xB by simply letting
t per unit be equal to the green vertical distance in the graph. As a result, buyers in the market
would face the higher price pB while sellers would receive the lower price pS with the difference
between the two prices representing the payment t per unit in taxes. A tax such as this that is
intended to reduce market output to its efficient quantity because of the presence of a negative
production externality is called a Pigouvian Tax.2
In panel (b) of the Graph we can then analyze more directly how this tax is efficient. In the
absence of the tax, the market produces output xM at price pM . You can check for yourself, in
a way exactly analogous to what we did in panel (c) of Graph 21.1, that the competitive market
on its own will produce overall surplus equal to (a + b + e + h − j), with the triangle j once
again representing dead weight loss. Under the tax t, however, consumer surplus (a) and producer
surplus (h + i) combine with a positive tax revenue (b + c + e + f ) and a social cost from pollution
(c + f + i) to produce an overall surplus (a + b + e + h). This is exactly equal to the green maximum
surplus achieved by benevolent Barney in Graph 21.1b — and eliminates the deadweight loss (j).
Put differently, the reason we found taxes to be inefficient in Chapter 19 was that they distorted
the price signal that coordinated efficient cooperation between producers and consumers – but, in
2 The tax is named after Arthur Cecil Pigou (1877-1959), a British economist and student of Alfred Marshall (who
succeeded Marshall as Professor of Political Economy at Cambridge University). Pigou developed the distinction
between private and social marginal cost in his most influential work entitled Wealth and Welfare.
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763
Graph 21.2: An Efficient Pigouvian Tax
the presence of externalities, the price signal is already distorted insofar as it does not efficiently
coordinate production and consumption. The tax then removes the distortion and causes the market
to “internalize the externality”.
In order for the government to be able to impose an efficient Pigouvian tax t, it must however
know the optimal quantity xB it wants the market to reach and it must know the difference between
the market demand and supply curve at that quantity. Put differently, the government must
know the marginal social damage caused by pollution at the optimum quantity. If it possesses this
information, the government can achieve the maximum social surplus by simply setting the per-unit
tax equal to this marginal social damage of pollution.
Exercise 21A.3 What if the government only knows the marginal social damage of pollution at the equilibrium output level xM and sets the tax rate equal to this quantity? Will this result in the optimal quantity
being produced? If not, how do the the SM C and the supply curve have to be related to one another in
order for this method of setting the tax to work?
It may in principle not look too difficult for the government to gather sufficient information
to implement a Pigouvian tax that causes markets to once again produce efficiently. However,
suppose that there are now many different industries, each causing pollution. In order to set optimal
Pigouvian taxes, the government now has to know this same information for each industry and set
the per unit tax in each industry, letting taxes vary across polluting industries as the marginal social
damage of pollution at the optimum is different everywhere. This would then result in a complex
system of different Pigouvian taxes across all polluting industries. As technology changes, these
rates would have to be continuously adjusted. And, perhaps worst of all, unless the government
adjusts Pigouvian taxes whenever firms find ways of reducing pollution on their own, individual
firms in each industry would gain no benefit from applying pollution-abating technologies in their
own firms — because they would still face the same taxes. Thus, while it may look easy in principle
to impose Pigouvian taxes, it is much more difficult to do so in practice and to simultaneously
encourage those industries for whom it is easy to reduce pollution to do so in ways other than
simply cutting production due to the tax.
It is for this reason that economists have largely turned away from recommending Pigouvian
taxes on output and have instead turned to alternatives that focus more directly on forcing pro-
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Chapter 21. Externalities in Competitive Markets
ducers to confront the tradeoff between reducing pollution (through less production or through the
development of pollution abating technologies) or paying for its social costs. This shift in focus
has also been made possible by new technologies that allow governments to pinpoint who is producing pollution – and thus to require polluters to pay for pollution directly. This can be done
either through a pollution tax (as opposed to a Pigouvian tax on output), or through the design of
market-based environmental policy. We will discuss the latter first and then briefly compare it to
the former.
Exercise 21A.4 In Chapter 18, we discussed the efficiency losses from government mandated price ceilings
or price floors. Could either of these policies be efficiency enhancing in the presence of pollution externalities
(assuming the government has sufficient information to implement these policies)?
21A.1.3
Market-based Environmental Policy
The most common market-based environmental policy works as follows: The government determines
an overall level of pollution (of each kind) that it finds acceptable and then issues pieces of paper
that permit the owner to emit a certain quantity of different types of pollutants per week (or month
or year). These pieces of paper, known as pollution vouchers or tradable pollution permits, thus
represent the “right to pollute” by some amount. Then the government releases these rights —
either by auctioning them off or by simply giving them to different firms in different industries.
It turns out that it does not matter which precise way the government uses to distribute such
permits — the important feature for our analysis is that individuals who own such permits can
sell them to others if they so choose (and thus transfer the “right to pollute” to someone who is
willing to pay more than it is worth to the original owner). In essence, the policy therefore “caps”
the overall pollution level by fixing the number of pollution permits – and then allows “trade” in
permits to determine who uses them. For this reason, it has come to be known as a cap-and-trade
policy.
Pollution vouchers have value to producers because they permit producers to emit pollution in
their production process. At the same time, whenever a producer chooses to use such a voucher,
she incurs an economic (or opportunity) cost — because she could have chosen to sell (or rent) the
voucher to someone else instead. Each producer therefore has to weigh the costs and benefits of
using a pollution voucher — and each producer knows that she will have to use fewer vouchers the
less she produces and the more she takes advantage of pollution-abating technologies. Since some
production processes lend themselves to pollution-abating technologies more easily than others,
firms in some industries will have a greater demand for such vouchers than firms in other industries.
As a result, by introducing pollution vouchers into an economy (and prohibiting the emission of
pollution when firms do not own such vouchers), the government has created a new market — the
market for pollution vouchers.
Exercise 21A.5 Explain how firms face a cost for pollution regardless of whether the government gives
them tradable pollution vouchers or whether firms have to purchase these.
This market is depicted in Graph 21.3 where pollution vouchers appear on the horizontal axis
and the price per voucher appears on the vertical. By introducing only a limited quantity of such
vouchers, the government has set a perfectly inelastic supply at precisely that quantity which results
in the level of overall pollution across all industries. Firms that emit pollution in their production
processes are the demanders of such vouchers, with demand depending on how much pollution is
involved in producing different types of goods and how easy it is for firms to find ways of reducing
21A. The Problem of Externalities
765
the pollution emitted in production. Put differently, those firms that find it difficult to reduce their
pollution will be willing to pay more for the right to pollute than those who can easily put a filter
on their smokestacks. In equilibrium, pollution vouchers will then sell at price p∗ .
Graph 21.3: A Market for Pollution Vouchers
Assuming the government can monitor polluting industries effectively (which is becoming increasingly easy as pollution monitors are widely distributed by the Environmental Protection
Agency across different regions and as satellite technology is becoming increasingly effective at
detecting pollution emissions from very precise locations), a system of pollution vouchers then
achieves the following: First, it imposes a cost on polluters by requiring that they purchase sufficient pollution rights for the pollution they emit. This, then, causes an upward shift in firm M C
curves as pollution vouchers become an input into the production process, and with it a shift in the
market supply curve in polluting industries. Such a shift will result in less production of output
in such polluting industries. Second, the system introduces an incentive for firms to search for
(and invest in) pollution-abating technologies. So long as it costs less to reduce pollution from my
firm than the pollution vouchers would cost me, I now have an incentive to reduce my pollution
emissions. Third, the system creates an incentive for new firms to arise and to independently invest
in research and development of pollution-abating technologies because the system has increased the
demand for such technologies in light of the fact that polluters would otherwise have to pay for
vouchers in order to produce.
As a result, the system achieves an overall reduction in pollution at the least social cost and
without the government adjusting any policy to changing conditions. The government does not have
to be in the business of picking which industry reduces which type of pollution by how much, and
it does not have to adjust those policies as pollution-abating technologies (that are more applicable
to some industries than to others) are produced. All the government has to do is to set an overall
pollution target and print a corresponding quantity of pollution vouchers. The newly created
pollution voucher market then rations who gets the vouchers and who does not get them — with
those for whom reductions in pollution are most costly choosing to use vouchers and others choosing
to reduce pollution cheaply. Put differently, pollution vouchers are government interventions that
harness the power of a newly created market to generate the information required to reduce pollution
at the lowest possible cost without any further government interference.
Exercise 21A.6 If the government, after creating the pollution voucher market, decides to tax the sale of
pollution vouchers, will there be any further reduction in pollution? (Hint: The answer is no.)
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Chapter 21. Externalities in Competitive Markets
And there is one final check on the system: While we have said thus far that polluters are the
ones who will form the demand curve in the pollution voucher market, it is in principle possible
to allow anyone at all to participate in that market. If, for instance, a group of deeply concerned
citizens feels that the government is permitting too much pollution to be emitted into the air, they
could pool resources and purchase some quantity of the vouchers — thus increasing the price (and
raising the cost to polluting) while lowering the supply (if they simply store away the pollution
vouchers). As we will see in a later chapter on public goods, such groups face a difficult free rider
problem that they need to overcome, but if they can, they are able to impact the overall level of
pollution without lobbying the government.
One last clarifying caveat, however: While pollution vouchers offer a mechanism to reduce
pollution to a target level in the least costly way, there is nothing in a pollution voucher system
that guarantees we will have set the socially optimal target for pollution to begin with. If the
political process that determines this target is efficient, then the target will be set optimally. But
otherwise, the target might be too high or too low – all that the cap-and-trade system does for us
is to get us to the target in the least costly way.
21A.1.4
Pollution Taxes, Pigouvian Taxes and Cap-and-Trade
While the idea of taxing output in polluting industries – as originally proposed by Pigou – has lost
considerable favor among economists, the very technology that allows the establishment of markets
in tradable pollution permits now enables governments to tax pollution (rather than output) directly.
One suspects that, had Pigou thought it possible to detect pollution where it is emitted, he would
most likely have favored taxing pollution rather than output as well. Taxing pollution directly has
the same advantages over Pigouvian taxes that we have pointed out for cap-and-trade systems,
and a per-unit-of-pollution tax is in fact equivalent to establishing tradable pollution permits if the
tax rate is set at the same level as the price-per-unit-of-pollution that emerges in cap-and-trade
systems. Both systems provide incentives for firms to invest in pollution abating technologies;
neither requires governments to adjust industry tax rates as circumstances change (as it the case
under Pigouvian taxes on output); overall pollution is reduced in the least cost ways as firms for
whom it is easy to reduce pollution will do so rather than incur the cost of pollution (by either
paying a pollution tax or using pollution vouchers); and neither system automatically results in
full efficiency unless the government has lots of information on what the efficient tax rate or the
efficient number of pollution permits is.
Exercise 21A.7 In one of the 2008 Presidential Primary debates, one candidate advocated the cap-andtrade system over a carbon tax on the grounds that the carbon tax would be partially passed onto consumers
in the form of higher prices. Another candidate who also supported the cap-and-trade system corrected this
assertion – suggesting that, to whatever extent a carbon tax would be passed onto consumers, the same is
true of costs (of tradable permits) under the cap-and trade system. Who was right?3
While pollution taxes and cap-and-trade systems are therefore quite similar, environmental
policy makers nevertheless debate their relative merits. Some consider it important to set precise
target levels for pollution – with cap-and-trade systems allowing an easy way of establishing such
targets while then letting the market for tradable permits determine the per-unit-of-pollution price
required to implement the target. Others believe it is more important to specify the per-unitof-pollution cost directly through a tax in order to allow firms to plan accordingly – leaving the
3 The exchange took place in the January 5, 2008 Democratic Presidential Primary Debate held at St. Anselm
College. The first candidate was New Mexico Governor Bill Richardson; the second was then-Senator Barak Obama.
21A. The Problem of Externalities
767
level of pollution reduction that result to arise from firm responses to the tax. Again, if the perunit-of-pollution tax is set at the same rate as the per-unit-of-pollution price that emerges under
a particular “cap” in a cap-and-trade system, the two policies have identical effects – but one gets
there by being precise about the target pollution level up front while the other gets there by being
precise about the per-unit-pollution cost up front.
A second issue that is raised in policy debates regarding cap-and-trade versus pollution taxes
relates to politics and implementation. Some fear that a nation-wide – or even world-wide – cap-andtrade system would involve excessive government bureaucracy to administer the various markets for
different types of pollution vouchers while others argue that administering pollution taxes would
involve similar issues. In practice, however, there appears to be one important political reason for
environmental policy makers to favor the cap-and-trade system: It has a built in mechanism for
overcoming concentrated opposition from industries that are particularly affected. Such industries
would face increased marginal costs under both the pollution tax and the cap-and-trade system, but
pollution vouchers could be given away for free to some industries in order to “buy” their political
support. In essence, this involves a transfer of wealth (in the form of pollution vouchers that can be
traded) without a change in the increased opportunity cost of emitting pollution. Under pollution
taxes, one could similarly “buy off” industry opposition through transfers of taxpayer money, but
this appears to be politically more controversial.
Exercise 21A.8 Suppose that advocates of pollution taxes proposed a reduction in such taxes for key industries that would otherwise be opposed to the policy. How is this different than giving pollution vouchers
away for free to such key industries in a cap-and-trade system?
Finally, to the extent to which the pollution problem to be addressed is global (as in the case of
greenhouse gases) rather than local (as in the case of acid rain), policy makers may favor the capand-trade system as it permits the establishment of global markets in tradable pollution permits
to achieve global reductions in pollution while allowing an initial establishment of country-specific
“caps” through negotiated international agreements. Such a system does not enshrine countryspecific caps because permits could be traded across national boundaries, but – much as support
form particular industries can be gained by giving some pollution permits away – international
support for such agreements could be facilitated by initially allocating relatively more pollution
permits to some countries rather than other countries.
Exercise 21A.9 Less developed countries often point out that countries like the U.S. did not have to
confront the fact that they caused a great deal of pollution during their periods of development – and thus
suggest that developed countries should disproportionately incur the cost of reducing world-wide pollution
now. Can you suggest a way for this to be incorporated into a global cap-and-trade system?
21A.2
Consumption Externalities
We have thus far considered only externalities generated in the production of goods and, with the
exception of the externality considered in within-chapter-exercise 21A.1, we have limited ourselves
to externalities that have negative impacts on others — or what we have referred to as negative
externalities. Externalities can, however, arise in production and consumption, and they can be
positive or negative. We will now illustrate the impact of an externality on the consumer side, and,
to differentiate it further from what we have done so far, we will consider a positive rather than a
negative externality.
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Chapter 21. Externalities in Competitive Markets
Suppose, for instance, that production of hero cards entails no pollution whatsoever but, whenever a consumer purchases hero cards for children, the world becomes a better place. In particular,
suppose that for each child that is exposed to hero cards, future crime falls and good citizens emerge.
This may sound silly because of the context of the example, but such arguments are often made
in markets like children’s programming on television or markets involving the arts. The essential
nature of the argument is always the same: In addition to the private benefits that consumers
obtain directly from consumption, others in society benefit indirectly in ways that are not priced
by the market.
21A.2.1
Positive Externalities from Consumption
Graph 21.4 then presents a series of graphs for positive externalities that is exactly analogous to
the series of graphs in Graph 21.1 for negative externalities. Panel (a) simply illustrates consumer
and producer surplus along market supply and demand curves (once again under the assumption
that demand can be interpreted as marginal willingness to pay). Panel (b) introduces a new curve
called ”SM B” or social marginal benefit. This curve includes all the benefits society gains from
each unit of consumption. It therefore includes all the private benefits that consumers get (and
that are measured by the demand curve), plus it includes additional social benefits that are gained
by others. As in the case of SM C and supply, SM B and demand can be related to each other in a
variety of ways, but under positive externalities SM B must certainly lie above demand (or private
marginal willingness to may).
Graph 21.4: Underproduction in the Presence of a Positive Externality
Our benevolent social planner would then use this SM B to measure the marginal benefit of
each hero card that is produced (while measuring the marginal cost along the supply curve in the
absence of negative externalities.) He would therefore choose the production level xB in panel (b)
of the graph, giving the shaded green area as overall social surplus. Thus, the market produces an
inefficiently low quantity of a good that exhibits a positive consumption externality. We can derive
the exact deadweight loss from the areas labeled in panel (c) of Graph 21.4. At the competitive
market equilibrium, consumer surplus is simply area (a) (equivalent to the blue area in panel (a))
and producer surplus is area (b) (equivalent to the magenta area in panel (a)). Since the market
21A. The Problem of Externalities
769
produces an output level xM , the additional social benefit from the externality is given by area (c).
Thus, the market achieves an overall social gain equal to area (a + b + c). Our social planner, on
the other hand, achieves that plus area (d) — implying that society incurs a deadweight loss of (d)
in the absence of non-market institutions that induce additional production.
21A.2.2
Pigouvian Subsidies
One non-market institution that we already know from our previous work can raise the level of
output in the market is a government subsidy. Suppose that the government knows it wants to raise
output in the hero card market to xB above the market quantity xM . In panel (a) of Graph 21.5,
this implies that the government can accomplish its goal by imposing a subsidy s equal to the green
vertical distance, thus lowering the price for buyers to pB and raising the price for sellers to pS .
Our discussion of the economic incidence of a subsidy in Chapter 18 treats this in more detail and
illustrates that the degree to which prices faced by buyers and sellers change depends on the relative
price elasticities of market demand and supply curves. When such a subsidy is used to “internalize
a positive externality”, it is known as a Pigouvian subsidy. As in the case of a Pigouvian tax, it
can restore efficiency by removing the externality-induced distortion in market prices.
Graph 21.5: An Efficient Pigouvian Subsidy
Suppose again (for simplicity) that tastes for hero cards are quasilinear and that we can therefore
treat the market demand curve as the aggregate marginal willingness to pay curve for consumers.
In panel (b) of the graph we can then calculate the areas that make up total surplus before and after
the subsidy. Before the subsidy, consumer and producer surplus simply sum to (a + b + c + d) and
non-market participants gain additional surplus of (e + f ). Thus, total surplus under pure market
allocations is (a+b+c+d+e+f ). Under the subsidy, consumer surplus is (a+b+c+g +k), producer
surplus is (b+c+d+f +i) and surplus for non-market participants is (e+f +h+i+j). From the sum
of these areas we then need to subtract the cost of the subsidy, which is (b + c + f + g + i + j + k) —
giving us a total surplus of (a + b + c + d + e + f + h + i). Thus, total surplus under the subsidy
is now equal to the green area in Graph 21.4b which we concluded was the maximum social gain
possible, with the subsidy having eliminated the deadweight loss (h + i) that occurred under a pure
market allocation.
Exercise 21A.10 Suppose that, instead of generating positive consumption externalities, hero cards actually
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Chapter 21. Externalities in Competitive Markets
divert the attention of children from studying and thus impose negative consumption externalities. Can you
see how such externalities can be modeled exactly like negative production externalities?
21A.2.3
Charitable Giving, Government Policy and Civil Society
In the case of a negative production externality of pollution, we illustrated next how government
could, instead of attempting to calculate all the “right” Pigouvian taxes each year, create a new
market of pollution vouchers that can efficiently reduce pollution to some level set by the government. In the case of positive consumption externalities, I can’t offer a similar market-based
policy that is currently under discussion, but we should note that the market outcome we have
predicted in the model may not necessarily be the actual outcome if markets operate within the
context of non-governmental and non-market institutions that we referred to in Chapter 1 as civil
society. The word “civil society” does not have a clear definition and is often used to mean many
different things. In this text, I will refer to an institution as a “civil society” institution whenever
it is not clearly set up by the government nor does it operate strictly on the self-interested motives
that generate explicit prices in markets. Civil society institutions are then the sets of interactions
among individuals that occur outside the context of government and outside the context of explicit
market prices. Such institutions tend to arise as individuals try to use persuasion rather than the
political process to address issues of concern that are not addressed in the market. The existence of
positive consumption externalities offers an example — because, as we have seen, it is a case when
the market in the absence of non-market institutions produces too little of goods that are valued
in society beyond their simple consumption value.
As you are no doubt aware, many organizations spend substantial energy in attempts to make
people aware of many social concerns in an attempt to persuade them to voluntarily contribute
money or time to organized efforts aimed at addressing such concerns. In the case of television
programming for children, for instance, we have all seen appeals on television for private donations
to increase funding for such programs. Such efforts to appeal for charitable donations run into
difficulties involving “free riding” that we will address more explicitly in Chapter 27 and thus offer
no guarantee of achieving a fully efficient outcome, but they appear to play an important role in
many circumstances where positive externalities would make markets by themselves produce too
little.
At this point, we will simply leave the issue with the observation that all three types of institutions that we have discussed — government, markets and civil society, face obstacles in achieving
efficient outcomes. Markets, as we have seen, will tend to underproduce in the presence of positive externalities and overproduce in the presence of negative externalities; governments may face
difficulties in ascertaining the information necessary for implementing optimal outcomes through
taxes, subsidies (or other means), especially as circumstances within societies change, and they face
political hurdles that we will treat more explicitly in Chapter 28. And civil society efforts that
rely on strictly voluntary engagement of non-market participants face difficulties in engaging those
non-market participants fully as each will tend to rely on others to address the problem. Yet each
appears to play a role in the real world.
Finally, just as the case of pollution vouchers represents an effort by government to engage
market forces in finding efficient solutions to excessive pollution, government policies are often
aimed at engaging civil society institutions more. The most obvious example of this can be found in
the U.S. income tax code that offers tax deductions to individuals who voluntarily give to charitable
causes — thus subsidizing such causes without the government making the explicit decision of which
21A. The Problem of Externalities
771
charities will end up engaging non-market participants. Thus, when the government faces too many
hurdles in designing explicit subsidies for each industry that generates positive externalities, it can
offer such general subsidies aimed at reducing the hurdles faced by civil society organizations in
finding non-market, non-governmental solutions.
Exercise 21A.11 In what sense does the tax-deducibility of charitable contributions represent another way
of subsidizing charities?
Exercise 21A.12 In a progressive income tax system (with marginal tax rates increasing as income rises),
are charities valued by high income people implicitly favored over charities valued by low income people?
Would the same be true if everyone could take a tax credit equal to some fraction k of their charitable
contributions?
Exercise 21A.13 We did not explicitly discuss a role for civil society institutions in correcting market
failures due to negative externalities. Can you think of any examples of such efforts in the real world?
21A.3
Externalities: Market Failure or Failure of Markets to Exist?
Thus far, we have seen that markets by themselves will produce inefficient quantities of goods
that exhibit positive or negative consumption and production externalities. In the absence of
government intervention, civil society efforts may contribute to greater efficiency. Alternatively,
government policies can be designed to change market output directly (as in the case of Pigouvian
taxes and subsidies) or to indirectly harness the advantages of market forces (as in the case of capand-trade policies) or civil society institutions (as in the case of the tax deductibility of charitable
contributions) to increase efficiency and lower deadweight losses. After we have explored more
fully (in the upcoming chapters) the many hurdles faced by markets, governments and civil society
institutions in implementing optimal outcomes for society, we will return in the last part of the
book to a general model of how we can ascertain the appropriate balance of markets, government
and civil society depending on the particulars of the social problem that is to be solved.
In the meantime, however, we can see yet another efficiency-enhancing policy tool the government has at its disposal by exploring a little more deeply the fundamental problem created by the
presence of externalities. We have seen that markets by themselves will tend to “fail” in the presence of externalities — and this has often led economists to refer to externalities as one (of several)
potential market failures. In this section, we will see how this market failure arises because of the
fact that, whenever there is an externality generated in competitive markets, we can trace the over
or under production that arises from this externality to the lack of a market or the non-existence of
a market somewhere else.
21A.3.1
Pollution and Missing Markets
Consider again the case of a market in which pollution is a by-product of production. The fundamental reason that a market will overproduce in this case (relative to the efficient quantity) is that
producers are not forced to face the full costs they impose on societies when making production
decisions. In particular, if the pollution that is generated is air pollution, the producer escapes
paying for the input “clean air” that is used in the production process unless some mechanism (like
Pigouvian taxes, pollution taxes or pollution vouchers) is implemented. Were there a market for
each of the inputs used in production — including the input “clean air”, the producers would have
to fully pay for all the costs they impose. Air pollution therefore arises as a problem that keeps
markets from producing efficiently because one of the inputs into production is not bought and sold.
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I know that this sounds rather silly — how could there possibly be a “market for clean air” when
no one owns the air and therefore no one can sell clean air to firms that use it in the production
process. It sounds silly because it is silly. Nevertheless, if we can suspend disbelief for a moment, we
can see the conceptual point that the externality is a problem precisely because we have not found a
way to create a market in clean air. If there was such a market, and if all air was owned by different
people, then each user of clean air would have to pay for it as it is being used. Consumers of clean air
— including producers who use clean air as an input — would have to pay for clean air just as firms
have to pay for labor and capital. Such a market for clean air would therefore result in a market
price which would, in the absence of any other externalities, result in maximum social surplus in the
clean air market. As producers contemplate production that involves pollution, they would then
face a price for clean air — shifting their marginal cost curves up and thus shifting market supply
up to be equal to the social marginal cost (SM C) of production rather then the marginal private
cost that excludes the social cost of pollution. This would then result in the efficient quantity of the
pollution-generating output, with social surplus once again maximized purely by market forces.4
In an abstract conceptual sense, the market failure generated by the presence of externalities
can then be traced to the failure of a market to exist. Does recognizing this get us any closer to
solving the problem? In the case of pollution, it is that recognition that has led economists to come
up with the proposal for creating markets in pollution vouchers. Pollution voucher markets are not
the same as markets in clean air, but they represent an attempt to resolve a problem created by
the non-existence of a market (for clean air) through the creation of a different type of market that
can help. Recognizing the market failure generated by externalities as a failure of a market to exist
can therefore create the opportunity for innovative government interventions that may, at least in
some cases, work better than other government solutions we might otherwise implement.
21A.3.2
The Tragedy of the Commons
This insight then points toward a huge role that governments more generally have to play in
order for markets to function efficiently. Throughout our treatment of the efficiency of markets in
Chapters 15, 16 and 17, for instance, we made the implicit assumption that markets for all sorts of
inputs such as labor and capital actually exist. Presuming that such markets exists presumes that
individuals own resources that they can trade — and this presumes that there is some mechanism in
place that protects the property rights of owners of resources. Firms cannot just take my leisure and
use it for labor inputs — they are required to persuade me to sell my leisure to them by offering
me a wage that I consider sufficient. Similarly, they cannot just take my savings or retirement
account and use the money to buy labor, land and equipment — they have to pay for using my
financial capital by paying me interest. All this requires a well-established system of legally enforced
property rights, and such a system has in practice typically required government protection and a
well-functioning court system to enforce property rights.5
Externalities, as we have seen, arise when such property rights have not been established. Pollution is a problem because there does not exist a system of property rights to clean air that forces
4 It is noteworthy that it does not actually matter who owns the right to clean air — whether individuals or firms
own this right, a market that prices the use of clean air in production would form. If the polluter herself owns the
right to the air, she is still facing the cost of polluting because her opportunity cost of using the clean air in her own
production is to sell the clean air to someone else in the market. We will say more on this later on.
5 Most of us, including me, take for granted that such protection of property rights must be provided by government.
And it usually is. But there are contrarian voices among some economists and philosophers that maintain government
is not necessary for protection of private property to emerge. We will say a bit more about this in Chapter 30.
21A. The Problem of Externalities
773
firms to pay for using clean air as an input into production. In effect, without some other institution
in place, firms are simply able to take clean air for free as they produce goods — something we
do not permit for inputs like labor and capital. Were they to similarly be able to take my leisure
and capital — were there no legal system of property rights in those input markets, we would have
even worse externality issues to deal with. Whenever a resource is not clearly owned by someone, it
therefore becomes possible for economic agents to take those resources without incurring a cost —
even though this imposes costs on society. It is then a logical consequence that, if it is feasible for
the government to establish a system of property rights in resources that are not currently owned
by anyone, such government interference can create additional markets that reduce the problem of
externalities by forcing market participants to face the true social cost of what they are doing.
For this reason, economists have come to refer to externality problems that arise from the nonexistence of markets as the “Tragedy of the Commons” — the “tragedy” of social losses that emerges
when property is “commonly” rather than privately owned. We could say, for instance, that clean
air is owned by everyone, but that simply means it is owned by no one in particular. Parents know
this tragedy well. When we give toys to our children as common property to be shared without
any guidance or rules, our children tend to fight like cats and dogs as they try to get those toys for
themselves. Most parents therefore quickly learn that conflict is reduced if clear ownership of toys
is established, with each child knowing (to the extent that children fully internalize this) that they
have to get permission from the other child when seeking to use that child’s toys. When parents
realize this, they act as economists who understand the tragedy of the commons.
More generally, much human suffering in the world can be directly traced to societies not heeding
the lessons of the Tragedy of the Commons. Entire societies have been set up in attempts to abolish
private property and replace the mechanism of markets with some alternative mechanism. It takes
only a quick glance at 20th century history, for instance, to see how much societies that have
protected private property and (thus established markets) have economically thrived while societies
that have attempted to do the opposite have failed. A full understanding of externalities suggests
that such societies failed because they created huge externalities by eliminating markets without
finding an alternative government or civil society mechanism to generate social surplus. In short,
by not supporting markets, they have created large “tragedies of the commons.”
Exercise 21A.14 Large portions of the world’s forests are publicly owned – and not protected from exploitation. Identify the tragedy of the commons – and the externalities associated with it – that this creates.
Exercise 21A.15 Why do you think there is a problem of over-fishing in the world’s oceans?
21A.3.3
Congestion on Roads
We do not, however, have to dig into historical examples of non-market based societies or reach for
the pie in the sky of “markets in clean air” to see the relevance of an understanding of the Tragedy of
the Commons in thinking about solutions to externality problems. Economists who have estimated
the social cost of externalities in the U.S., for instance, have found that the social cost of time
wasted on congested roads rivals the social cost of environmental damage from pollution. Think
of your own experiences being stuck in traffic — it is mind-numbing to be stuck in traffic even for
short periods of time because the opportunity cost of our time is large. In some of our larger cities,
commuters routinely spend significant amounts of time in precisely such a position.
The problem of congested roads is an example of a Tragedy of the Commons. Roads, by and
large, are commonly or publicly owned — which is to say that they are not owned by anyone. As
you and I get on the road, we may think about the cost of taking the drive into the city — the
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cost of our time, the gasoline we use and the depreciation of our car. We do not, however, think
about the cost we are imposing on everyone else that is also taking a trip. Put differently, there
is a negative externality each of us imposes on everyone else who is on the road at the same time
as we add to the congestion of the road. In the absence of a mechanism that makes us face this
social cost of our private actions, we therefore will tend to take too many trips – and we will be on
the road at the “wrong” times. You may say that surely my own contribution to the congestion of
the roads is minor, but all of us together are causing the congestion problem that wastes billions
of dollars worth of time each week on the congested roads of larger cities. If my entry onto the
road causes thousands of others to take even one more second to get to where they are going, I am
imposing quite a social cost on others without paying any attention to it.
Exercise 21A.16 Can you think of an other costs that we do not think about as we decide to get onto
public roads?
Solutions for this particular Tragedy of the Commons are still evolving, and changes in technology are playing a large part in shaping these solutions just as new technologies that permit
detection of pollution have shaped new environmental policies (such as pollution taxes and capand-trade systems). The difficulty in finding a way for individuals to internalize the social cost
they are imposing on others on the road lies in the difficulty of establishing a market that will price
that social cost. In the past, economists have often proposed somewhat blunt policies falling into
two general categories: First, we can impose a tax on gasoline that will raise the cost of driving
and therefore reduce the amount of driving individuals will undertake; and second, when there are
sufficiently many individuals in sufficiently dense geographic areas, governments can design public
transportation systems like subways that are expensive to build but that, once built, can offer
attractive alternative means of transportations within cities.
The building of public transportation may alleviate congestion, but it does not in itself address
the Tragedy of the Commons that remains on public roads — and it may create a different Tragedy
of the Commons if public transportation is priced in such a way as to cause congestion in buses,
subways and so forth. Nevertheless, it has represented an important element of addressing crowding
on roads in some urban areas. Taxation of gasoline is appealing in that it does raise the cost
of driving and bring it more into line with the social cost of individual decisions during peak
traffic hours, but it also raises the cost of driving during non-peak hours when congestion is not a
problem — thus creating deadweight losses during those hours just as it reduces deadweight losses
during peak hours.
Exercise 21A.17 Are there other externality-based reasons to tax gasoline?
In recent years, however, it has become possible to price driving on congested roads more directly
through tolls. Before the advent of electronic equipment that has made this easier, such tolls have
involved toll booths which themselves can contribute to congestion around the booths as traffic
slows down even as it keeps individuals off the roads. As technology improves, however, we are
beginning to see increasingly efficient mechanisms for tolls to be imposed, mechanisms that do not
require individuals to stop, reach into their wallets and pay a toll-booth attendant. As a result, we
are seeing cities increasingly use electronic tolls that can vary with the time of day that individuals
choose to use roads. User fees in the form of tolls then represent an attempt to make individuals
face the social cost of driving during peak hours. At least in principle, such technology also permits
the more direct establishment of markets in roads — markets in which road networks are privately
owned and the use of the road is priced within markets. As technology and our understanding of the
21A. The Problem of Externalities
775
underlying causes of externalities on roads is changing, we therefore see the emergence of new ways
for government policy to interact with markets to reduce the social costs of an important externality.
If such topics are of interest to you, you might consider taking an urban or transportation economics
course at some point.
Exercise 21A.18 Some have argued against using tolls to address the congestion externality on the grounds
that wealthier individuals will have no problem paying such tolls while the poor will. Is this a valid argument
against the efficiency of using tolls?
21A.4
Smaller Externalities, the Courts and the Coase Theorem
We have thus far focused primarily on externalities that affect many individuals – such as pollution
and congestion. But many of the externalities that we are most aware of in our daily lives are much
less grand – the loud music in the dorm-room next to yours, the odor from the student who insists
on sitting next to you in class but who also insists on showering infrequently, the insensitivity of
the person on the bus who appears to be talking loudly to himself but is actually speaking on his
well-hidden cell-phone, or that baby that just stopped screaming only to have switched from an
externality that affects the auditory nerve to one that affects our sense of smell. These are all
negative externalities – but we could think of positive ones as well. When I smile in the hallway at
work, a few people a day might derive direct benefits from my cheerful disposition, or when I open
the door for a student carrying heavy books (such as the one you are reading – sorry, I don’t know
how to be brief!), that student’s life might be just a bit better today – even more so if I happen at
the moment to be offering a rousing rendition of “O Sole Mio” . If you think about it, externalities
are everywhere that people operate within close proximity to one another – in the workplace, in
restaurants, in neighborhoods. And sometimes these externalities cause us to take each other to
court.
21A.4.1
The Case of the Shadow on your Swimming Pool
Consider, for instance, the following example: You and I live next to each other in peace and
harmony. Suddenly I win some money in the lottery and decide that I want to add to my house.
So I draw up some plans to add an additional floor to my existing house. Normally you would
not care about this, but it turns out that the additional floor will cast a long shadow onto your
property – and in particular the area of your property that currently contains a beautiful (and
sunny) swimming pool. You get very upset that your swimming pool will suddenly be in the shade
all the time, and so you go to court and ask the judge to stop my building plans. Your legitimate
argument is that I am imposing a negative externality that I am not taking into account. “He must
be stopped,” you insist to the judge.
The judge sees your point but he wants to be careful and is trying to figure out whether it would
or would not be efficient to build the addition to my house despite the adverse effect this will have
on you. Maybe I get a lot more enjoyment from the addition than you lose from the shade on your
swimming pool, or maybe it’s the other way around. Maybe it would cost you very little to move
to a different house and have someone that does not care about the shade on the swimming pool
move into your house (thereby eliminating the very externality we are worried about). Or maybe
it would be easy for me to find a bigger house elsewhere and relatively costless for me to move. It’s
hard to tell without the judge figuring out a lot of details about the case. And, one might argue,
that there isn’t an easy way to judge this on a basis other than efficiency. After all, we both are
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Chapter 21. Externalities in Competitive Markets
equally to blame for the existence of the externality – it would not exist if I were not trying to build
an addition, but it also would not exist if you were not so insistent on having the sun shining on
your silly pool!
21A.4.2
The Coase Theorem
Ronald Coase, an economist at the University of Chicago, came along and had a neat insight that
might, under certain conditions, make the judge’s life a lot easier.6 He thought that the reason you
are taking me to court is that we are confused about who has what “property rights” – and this
ambiguity is making it difficult for us to come up with the optimal solution to our problem on our
own. Suppose, for instance, you knew the judge would rule that I had the right to build regardless
of the damage this does to you. You might then invite me for coffee and ask if there isn’t a way
you could convince me to not build my addition. If the damage that is done to you is greater than
the pleasure I get from my addition – i.e. if it would be efficient for me not to build the addition –
you would in fact be willing to pay me an amount that will make me stop the addition. Perhaps
I would find another way to add to my house, or perhaps I would move with the money you gave
me to make me stop. If, on the other hand, your pain from the addition is less (in dollar terms)
than my pleasure – i.e. if it is efficient for me to go ahead with the addition – you would discover
over coffee that you aren’t willing to pay me enough to stop the addition. Perhaps you will just
stay and suffer in a shaded pool, or perhaps you’ll move elsewhere. But notice that, once you know
that I have every right to build the addition, you have an incentive to figure out whether you can
pay me to stop – and once you figure this out, you will insure that the efficient outcome happens.
The same is true in the case where I know that you have the “property right” – that is, that
you have a right to block my addition. In that case, I have an incentive to have you over for coffee
to my house to see if I could persuade you to let me go ahead. If the addition means more to me
than the pain it causes you, then you will be willing to accept a payment that I am willing to pay
in order to get you to drop your objections. If, on the other hand, my gain from the addition is less
than your pain, then I won’t be willing to pay you enough to get you to stop your objections. Thus,
if the initial “property right” rests with you, then I am the one who has an incentive to figure out
whether my gain is greater than your pain – and in the process get us to do what is efficient. Note
that neither one of us actually cares about efficiency – but, once we know who has what rights, our
private incentives make it in our interest to find the efficient outcome.
Exercise 21A.19 True or False: While it might not matter for efficiency which way the judge rules, you
and I nevertheless care about the outcome of his ruling.
To the extent to which we find this reasoning persuasive, Coase has just gotten the judge who
cares only about efficiency off the hook: No matter what the judge decides, you and I will arrive
at the efficient outcome – the most important thing is that he needs to define the property rights so
that we can have coffee and know what we are negotiating about. I know this problem well in my
house where I am frequently called upon to be the judge that adjudicates cases of property rights
disputes involving my two eight year old daughters. Knowing about Coase, I don’t even listen to
6 Ronald Coase (1910-), who won the 1991 Nobel Prize in part for his contribution to this area, has the rare quality
of being both an economist and a person so averse to math that it has been said of him (which is probably not true)
that he will not number the pages of his manuscripts. The article in which he put forth the Coase Theorem – “The
Problem of Social Cost,” The Journal of Law and Economics 3 (1960), 1-44 – is therefore quite readable by those
with math phobias – and incidentally is one of the most cited articles in all of economics.
21A. The Problem of Externalities
777
their arguments. I just flip a coin to decide who gets the property rights this time and then send
them off to negotiate with each other.7
21A.4.3
Bargaining, Transactions Costs and the Coase Theorem
The Coase Theorem then says it is essential that property rights be clearly defined in cases when
there are negative externalities but it is not necessarily essential how those rights are defined.
This should have a familiar ring – we just emphasized in the previous section that the absence of
“markets” for the externality is the real underlying problem with externalities. Coase’s argument
is similar, except that he does not insist that we have to have a competitive “market” in the
externality – all we need to do is establish who has what rights and then let people solve the
problem on their own by bargaining with one another. In our example of me building an addition
to my house that will then cast a shadow on your swimming pool, there is no hope of establishing
a real (competitive) market, but we can clarify property rights sufficiently to give us an incentive
to figure out how to solve the externality problem.
Coase was not, however, naive, and he recognized that there might be barriers that keep people
from getting together to bargain their way out of an externality problem once property rights are
fully defined. These barriers are called transactions costs, and if they are sufficiently high, you and
I might never have that coffee to talk about how to proceed. If we just can’t stand each other’s
presence in the same room, then there is a transactions cost to getting together – and when this is
the case, the judge’s decision suddenly matters a great deal more. If the efficient outcome is for me
to build my addition and the judge rules in your favor, these transactions costs would keep me from
getting together with you to offer you the payment necessary to let me proceed. Similarly, if the
efficient outcome is for me to not build the addition and the judge rules in my favor, transactions
costs again keep us from getting together in order for you to offer me the payment necessary not
to build. Thus, in the presence of sufficiently high transactions costs, the judge needs to figure out
what the efficient outcome is and then rule accordingly so that it is not necessary for us to get
together to solve the problem through side payments among each other. The full Coase Theorem
can then be stated as follows: In the presence of sufficiently low transactions costs, the efficient
outcome will arise in the presence of externalities so long as property rights are sufficiently clear.
We can then see that the Coase Theorem offers us a decentralized way out of externality problems so long as transactions costs are low, and transactions costs will tend to be lower the fewer
individuals are affected by an externality. If it’s just you and me arguing about whether I should or
should not build an addition that only affects the two of us, we might think that transactions costs
are in fact sufficiently low and we will bargain our way to a solution if the assignment of property
rights requires such bargaining. For this reason, we might not worry about all the every-day externalities that affect only small numbers of people – chances are probably better that individuals
themselves will figure out the efficient outcome than that a government with limited information
can dictate the efficient outcome. Put differently, as long as people in “small externality settings”
have reasonable expectations about how the law will treat externality issues if such issues were to
be adjudicated in a court room, such problems are best handled in the “civil society” in which
people interact voluntarily outside the usual price-governed market setting.
Exercise 21A.20 Use the Coase Theorem to explain why the government probably does not need to get
involved in the externality that arises when I play my radio sufficiently loud that my neighbors are adversely
affected, but it probably does need to get involved in addressing pollution that causes global warming.
7 My
wife thinks that makes me a bad parent. Weird.
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21A.4.4
Chapter 21. Externalities in Competitive Markets
Bees and Honey: The Role of Markets and Civil Society
The Coase Theorem applies to all types of externalities – positive or negative. So far, we have
been sticking with the example of the negative externality of the shadow cast on your pool by
the addition to my house. A classic example of positive externalities involves bee keepers and
apple orchard owners. It turns out, however, that – although the example was originally given as
motivation for Pigouvian subsidies, this is a case were Coase’s insights – as well as our more general
insights on markets and property rights – have held true in the real world, and there appears to be
no need for further Pigouvian interventions.8
Externalities in the case of bees and apple orchards abound. In order for apple trees to produce
fruit, bees need to travel from tree to tree to carry pollen from “male” to “female” trees. And in
order for bees to produce honey, they need some blossoms to visit. (You probably remember all this
from the “birds and the bees” talk that I recently had to have with my children.) Bee keepers that
let their bees roam therefore impose a positive externality on apple orchard growers (who benefit
from the cross-polination services), and apple orchard growers bestow a positive externality on bee
keepers (by providing them with the means for apple honey production). Even if we can figure out
a way for markets to solve this problem in general, there is the second problem: bees have a way
of not staying on the precise properties on which they were released. So if one orchard owner hires
cross-polination services (or invests in his own bees), the bees will cross into neighboring orchards
and provide services there – while also contributing to honey production.
In the absence of markets that can price all these externalities, our theory predicts that there
would be too few bees on apple orchards – resulting in too little cross-polination and too little
honey. As it turns out, however, none of this is a surprise to bee keepers and orchard growers.
Fairly sophisticated markets for bee-keepers to release their bees on orchards have emerged “spontaneously” – markets that established “themselves” in an environment where government’s only
role has been to guarantees the integrity of contracts and thus the property rights that are defined
in those contracts. The flowers on apple trees, it turns out, do not produce much honey — causing
the externality to go almost entirely from bee keepers to apple orchard growers. (The “apple honey”
that you can find on your supermarket shelves has precious little honey produced from apple trees
– it’s mainly the product of wild flowers that grow in the area of the orchards.) Clover, on the other
hand, produces tons of honey. Thus, growers of clover produce a net-positive externality for bee
keepers. While apple growers pay bee keepers to release their bees on the orchard, bee keepers pay
clover growers for permitting them to release their bees on the clover farms. This is an example of
competitive markets resolving an externality problem when property rights are well established.
Exercise 21A.21 In what sense do you think the relevant property rights in this case are in fact well
established?
This does not, however, resolve the more “local” externalities between orchard owners. If one
owner hires bee-services, those same bees cross over into other orchards – benefitting those growers
(while also benefitting the bee-keepers). Another economist has looked at this closely, and he
identifies a social custom that has emerged within the civil society – that is to say, outside the realm
8 This externality between bee keepers and orchard growers was pointed out by the economist James Meade (190795) who argued in 1952 that Pigouvian subsidies were needed to remedy the problem. Meade shared the Nobel Prize
in 1977 for contributions to the theory of international trade – which only goes to prove that even Nobel Laureates
can get it wrong (as Meade did in the case of subsidies in the bee keeping business.) To his credit, Meade wrote
eight years before Coase published his insights that came to be known as the Coase Theorem.
21B. The Mathematics of Externalities
779
of explicit market-based transactions and outside the realm of government intervention.9 This has
been dubbed the “custom of the orchards” – and it takes the form of an implicit understanding
among orchard owners in the same area that each owner will employ the same number of bee
hives per acre as the other owners in the area. While the Coase Theorem literally interpreted
suggests that individuals will resolve these “local” externalities through bargaining, this illustrates
another possible way for the theorem to unfold: Sometimes it is easier to converge on some local
understanding of appropriate behavior that can be sustained among small groups within the civil
society – rather than negotiate all the time about how many bee hives everyone is going to hire
this time around. (In part B of exercise 24.17, we investigate a game theoretic explanation for the
“custom of the orchards”).)
21B
The Mathematics of Externalities
We will begin our mathematical exploration of externalities in competitive markets (as in Section
A) with the motivating example of a polluting industry in partial equilibrium. Using linear supply
and demand curves, we can demonstrate how to calculate the optimal Pigouvian tax. Furthermore,
we will explore how the establishment of pollution permit markets can in principle achieve the
same efficiency gains as an optimally set Pigouvian tax and that, in fact, there exists a cap-andtrade policy that is equivalent to any Pivouvian tax policy in the absence of pollution abating
technologies. In the presence of such technologies, however, we will suggest, as we have in Section
A, that pollution voucher markets (as well as direct pollution taxes) have an inherent advantage
over Pigouvian taxes on output. While we won’t cover positive externalities (and accompanying
Pigouvian subsidies) in detail, the mathematics is virtually identical to that underlying Pigouvian
taxes and is therefore left as an end-of-chapter exercise.
We then turn to a more in-depth analysis of how externalities and the inefficiencies they give
rise to are fundamentally problems of missing markets. In particular, we’ll demonstrate how new
markets can be defined in an exchange economy that contains consumption externalities — and how
establishment of these new markets should in principle resolve the inefficiency from externalities.
We then demonstrate this in an extension of our example of a 2-person exchange economy from
Chapter 16 before concluding with a discussion of the Coase Theorem.
21B.1
Production Externalities
In the presence of sufficient information, it is not mathematically difficult to determine the extent
of deadweight losses from pollution or to arrive at an optimal Pigouvian tax. We will demonstrate
this here briefly with an example in which we use linear demand, supply and social marginal cost
SM C curves, and we will assume for convenience that the (uncompensated) market demand curve
is in fact also the appropriate marginal willingness to pay curve along which to measure consumer
surplus.
9 The economist is Steven Cheung – who was also the one who uncovered the contracts made by clover and apple
growers with bee keepers. This is discussed in considerably more detail in “The Fable of the Bees: An Economic
Investigation,” Journal of Law and Economics 17 (1973), 53-71.
780
21B.1.1
Chapter 21. Externalities in Competitive Markets
“Barney” versus the Market
We can begin with linear market demand and supply functions we have used in previous chapters,
with
xd =
B+p
A−p
and xs =
,
α
β
(21.1)
and we have previously calculated the competitive market equilibrium in this case as
pM =
A+B
βA − αB
and xM =
.
α+β
α+β
(21.2)
Now suppose that each unit of output x produces δ units of carbon dioxide pollution, and suppose
that the damage from this pollution increases quadratically with additional pollution dumped into
the air. In particular, suppose the externality cost is given by CE (x) = (δx)2 . Then the marginal
externality cost for each unit of x is the derivative of CE (x) with respect to x, or M CE = 2δ 2 x.
The inverse of the supply curve in equation (21.1) is the industry’s marginal cost curve — i.e.
M CS = −B + βx. Added together, these two curves make up the social marginal cost curve
SM C = −B + (β + 2δ 2 )x.
(21.3)
Exercise 21B.1 Suppose A = 1, 000, α = 1, β = 0.5, δ = 0.5 and B = 0. Illustrate the market demand
and supply as well as the SM C curves in a graph with x on the horizontal axis.
The efficient or optimal output level xopt — the output our mythical “Barney” would choose —
then occurs at the intersection of the SM C and the inverse demand curve p = A − αx. Solving the
equation −B + (β + 2δ 2 )x = A − αx for x, we get
xopt =
A+B
,
α + β + 2δ 2
(21.4)
which you can immediately see is less than the competitive equilibrium quantity xM in equation
(21.2).
Exercise 21B.2 Suppose the “pollution” emitted is actually not harmful and simply kills the mosquito
population in the area. The SM C of the pollution might then be negative — i.e. this kind of pollution might
actually produce social benefits. Will the efficient quantity now be greater or less than the market quantity?
Show this within the context of the example.
21B.1.2
The Efficient Pigouvian Tax
We can now determine the optimal Pigouvian tax topt that will insure that the market produces the
efficient level of output. In order for consumers to want to buy xopt , they must face a price pd such
that xopt = xd (pd ) = (A − pd )/α. Similarly, in order for producers to supply xM in equilibrium,
they must face a price ps such that xopt = xs (ps ) = (B + ps )/β. Solving these equations and
plugging in our solution for xopt from equation (21.4), we then get
pd =
(β + 2δ 2 )A − αB
βA − (α + 2δ 2 )B
and
p
=
,
s
α + β + 2δ 2
α + β + 2δ 2
(21.5)
21B. The Mathematics of Externalities
781
and subtracting ps from pd gives us the optimal Pigouvian tax topt required to get this difference
in consumer and producer prices; i.e.
topt = pd − ps =
2δ 2 (A + B)
.
α + β + 2δ 2
(21.6)
Exercise 21B.3 Complete exercise 21B.1 by illustrating and labeling the Pigouvian tax for this example.
Exercise 21B.4 Using the graph from the previous exercise, calculate consumer surplus, producer surplus,
the externality cost and overall surplus in the absence of the Pigouvian tax. Then calculate these again
under the Pigouvian tax, taking into account the tax revenue raised. What is the deadweight loss from not
having the Pigouvian tax?
21B.1.3
Cap-and-Trade
Now suppose that instead of imposing a tax t on output, the government requires that producers
hold a pollution voucher for each unit of carbon dioxide emitted in the production process. Since
every unit of output x produces δ units of pollution, a producer must therefore hold δ pollution
vouchers for every unit of output she produces. If the rental price of a voucher is r, this implies
that the industry marginal cost goes from (−B + βx) to
M CS = −B + βx + δr.
(21.7)
Exercise 21B.5 Illustrate how this shifts the supply curve in your graph (where you assume A = 1, 000,
α = 1, β = 0.5, δ = 0.5 and B = 0.).
Setting this equal to the (inverse) demand curve (which is p = A − αx) and solving for x, we
get the new equilibrium quantity (given a voucher rental rate of r) as
x∗ (r) =
A + B − δr
.
α+β
(21.8)
It is immediately clear from this equation, as it should be from the graph you drew in exercise
21B.5, that the market will produce less so long as the rental price of vouchers is greater than
zero. But this does not yet answer the question of how the rental price of vouchers is determined
in the first place. This price is, as demonstrated in Graph 21.3, determined in the new market
for pollution vouchers that the government creates when it limits the quantity of vouchers to some
level V .
Every unit of output causes δ in pollution – and thus requires δ pollution vouchers. In the
absence of any new introduction of pollution abating technologies, a total voucher level of V then
implies that the market will reduce its output to (1/δ)V . Substituting this into equation (21.8)
on the left hand side and solving for r, we get the equilibrium rental price for vouchers (given an
overall supply of vouchers fixed at V ) as
δ(A + B) − (α + β)V
.
(21.9)
δ2
Now notice what happens if the government provides exactly enough vouchers to allow the
market quantity xM = (A + B)/(α + β) to be produced; i.e. suppose the government sets V =
δ(A + B)/(α + β). Plugging this into equation (21.9), we get an equilibrium voucher price of zero —
r(V ) =
782
Chapter 21. Externalities in Competitive Markets
the voucher giving me the right to pollute ceases to be worth anything. For any level of V below
this, equation (21.9) tells us we will have a positive rental price for vouchers.
We can then relate this directly to Graph 21.3 in which we argued that firms will form a demand
curve in the market for vouchers while the government will set a perfectly inelastic supply by setting
a fixed voucher (and thus a fixed pollution) amount. In fact, equation (21.9) is the demand curve
(or the inverse demand function) for vouchers by polluting firms, and the inverse of this equation,
v(r) =
δ(A + B) − δ 2 r
.
α+β
(21.10)
is the demand function that relates the rental price r to the quantity demanded by producers.
Exercise 21B.6 Verify that a voucher price of zero results in the market output according to this demand
function.
Exercise 21B.7 Illustrate the demand curve for pollution vouchers and label its slope and intercept.
Here is an alternative (and perhaps more intuitive) derivation of the demand for pollution
vouchers: The maximum amount that a firm is willing to pay to be allowed to produce one more
unit of output depends on how much the firm thinks it can sell its output for and what the firm’s
other costs are. In panel (a) of Graph 21.6, for instance, if the government limited the quantity in
the market to x1 , the marginal firm would be willing to pay an amount equal to the blue distance
in order to produce — because this is the difference between the marginal cost of production for
this firm and the marginal willingness to pay for the output by the marginal consumer. Similarly,
if the government limited total output to x2 , the marginal firm would be willing to pay at most an
amount equal to the magenta distance.
Graph 21.6: Going from the Market for x to the Pollution Voucher Market
Now suppose we converted the units in which we measure x to voucher units — knowing that
we will have to have δ vouchers for 1 unit of output x. The marginal benefit (or inverse demand)
function when units are measured in terms of x is just the demand curve M B = A − αx. If we
now measure x in 1/δ-units, for instance, then the vertical intercept term of this function will have
to be divided by δ — giving a vertical intercept of A/δ for our new marginal benefit curve. The
21B. The Mathematics of Externalities
783
horizontal intercept of our original marginal benefit curve, on the other hand, has to change from
A/α to δA/α. From this, we can calculate the slope of our new marginal benefit curve as the
(negative) vertical intercept divided by the horizontal intercept, or (−(A/δ)/(δA/α)) = −α/δ 2 .
Thus, the marginal benefit curve when output is expressed in voucher-units is
δA − αv
α
A
,
(21.11)
− 2v =
δ
δ
δ2
and, applying similar logic to the producers’ marginal cost curve M C = −B + βx, the marginal
cost curve when output is expressed in voucher units is
M B(v) =
−δB + βv
.
(21.12)
δ2
Panel (b) of Graph 21.6 illustrates these marginal benefit and cost curves which are equivalent
to those in panel (a) except that the units on the horizontal axis are 1/δ the units in panel (a).
M C(v) =
Exercise 21B.8 What is the relationship between the length of the blue and magenta lines in panels (a)
and (b)?
Exercise 21B.9 Implicitly we are assuming δ = 2 in panel (b) of Graph 21.6. How would this graph change
if δ<1 — i.e. if each unit of output produces less than one unit of pollution?
The most that the marginal firm is willing to pay for a voucher is then simply the difference
between M B(v) and M C(v),
δ(A + B) − (α + β)v
,
(21.13)
δ2
exactly the expression for the voucher demand curve in equation (21.9). This function is graphed
in panel (c) of Graph 21.6, and the equilibrium price r∗ in the voucher market is then simply
determined by the intersection of this demand curve with the inelastic supply set at V by the
government or, mathematically, by substituting V for v in equation (21.13).
M B(v) − M C(v) =
Exercise 21B.10 If V = δx2 , which distance in panels (a) or (b) of Graph 21.6 is equal to r ∗ ?
Exercise 21B.11 For the case when A = 1, 000, α = 1, β = 0.5, δ = 0.5 and B = 0 (as you have assumed
in previous exercises), what is the rental rate of the pollution voucher when V = 250? What is the price of
a pollution voucher if the interest rate is 0.05?
21B.1.4
Pollution Vouchers versus Taxes
In Section A we drew a distinction between Pigouvian taxes (that are levied on output) and pollution
taxes which are levied on pollution that a firm emits. In our mathematical example here, there is a
one-to-one relationship between taxing output and taxing pollution – because we have assumed that
each unit of output produces δ units of pollution. Thus, in our simplified example, the Pigouvian
tax on output is not that different from a pollution tax – and, as a result, we can illustrate that,
for every pollution voucher cap under a cap-and-trade policy, there exists a tax that achieves the
same outcome. Keep in mind, however, that the real world introduces complexities that create a
real distinction between Pigouvian and pollution taxes, an issue we return to after demonstrating
the equivalence of tax and cap-and-trade policies for our example.
784
Chapter 21. Externalities in Competitive Markets
Equivalent Tax and Pollution Voucher
t
V
r∗ (V )
x xopt
$0 333
$0 667 500
$50 317
$100 633 500
$100 300
$200 600 500
$150 283
$300 567 500
$200 267
$400 533 500
$250 250
$500 500 500
$300 233
$600 467 500
$350 217
$700 433 500
$400 200
$800 400 500
$450 183
$900 367 500
$500 167 $1,000 333 500
Policies
DW L
$27,778
$17,778
$10,000
$4,444
$1,111
$0
$1,111
$4,444
$10,000
$17,778
$27,778
Table 21.1: A = 1, 000, α = 1, β = 0.5, δ = 0.5 and B = 0
Suppose the government knows the optimal level of output xopt in equation (21.4) as well as
the amount δ of pollution emitted by each unit of production. The information, combined with our
knowledge of supply and demand curves, is then sufficient to set the optimal voucher level at
V opt = δxopt =
δ(A + B)
.
α + β + 2δ 2
(21.14)
Plugging V opt into equation (21.9), this implies an equilibrium rental rate for vouchers of
r∗ (V opt ) =
2δ(A + B)
.
α + β + 2δ 2
(21.15)
In order to produce one unit of output, we have to rent δ vouchers – which implies that the
marginal cost of production has increased by
δr∗ (V opt ) =
2δ 2 (A + B)
.
α + β + 2δ 2
(21.16)
Note that this is exactly equal to the optimal Pigouvian tax topt we derived in equation (21.6);
i.e.
topt = δr∗ (V opt ).
(21.17)
Thus, so long as the government sets the number of pollution vouchers correctly, the market for
these vouchers will result in a price equal to the tax the government would have liked to impose
had it chosen to use a Pigouvian tax instead. In fact, as illustrated in Table 21.1 for the example
that you have worked with in many of the within-chapter exercises, for any tax imposed on outputs,
there exists and equivalent voucher level that will result in a voucher rental rate that has the same
impact on producers as the tax.
Exercise 21B.12 Suppose the government simply gives away the pollution vouchers. Why is the deadweight
loss the same under tax and cap-and-trade policies that satisfy t = δr ∗ (V ) (even though one makes revenue
for the government while the other does not)?
21B. The Mathematics of Externalities
785
Exercise 21B.13 Illustrate on a graph where the deadweight loss falls when t = 400 in Table 21.1. What
about when it falls at t = 100?
As noted above, however, our mathematical example obscures within its simplicity a difference
between taxing output in polluting industries and taxing pollution emissions directly. This is
because we illustrated the case of a single industry, ignoring the fact that many industries engage
in pollution – and we have not introduced the potential for pollution abating technologies to play a
role. Even within a single industry, a Pigouvian tax on output differs from a pollution tax in that
the latter allows firms to reduce their tax obligations by introducing pollution abating technologies
while the former does not. Thus, the equivalence of a Pigouvian tax to a pollution tax within an
industry only survives if we assume that the government will adjust the Pigouvian tax on output as
firms introduce pollution abating technologies. When considering pollution across industries, this
is further complicated by the fact that industries will differ in terms of the ease with which they
can introduce pollution abating technologies – with any equivalence between Pigouvian taxes and
pollution taxes then assuming that the government continuously adjusts Pivougian per-unit taxes
as pollution abating technologies are introduced in different settings. The equivalence between capand-trade and pollution taxes, on the other hand, is robust to the introduction of such real-world
complications.
21B.2
Consumption Externalities
The mathematics behind our graphical development of consumption externalities is almost identical
to that behind production externalities. As a result, we will treat this in end-of-chapter exercise
21.1 rather than developing it fully here. Instead, we will proceed next to considering the problem
of consumption externalities in a general equilibrium setting where we will be able to illustrate
more precisely what we mean when we say that the presence of an externality necessarily implies
the absence of a market that, if established, would eliminate the inefficiency that arises from the
externality.
21B.3
Externalities and Missing Markets
The idea of using (pollution voucher) markets to solve the externality problems created by pollution
is closely linked to a more general understanding of externalities as a problem of “missing markets”
(or, as we put it in Section A, of a failure of markets to exist.) The intuition behind this is not
difficult to see once we see how the missing markets could be defined — and how pricing within
those markets will then lead those who emit externalities to face the costs (or benefits) they impose
on others. Using our tools from Chapter 16, however, we can be a little more precise about what we
mean by missing markets and how an establishment of those markets resolves the inefficiency from
externalities under competition. We will do so below for the case of externalities in an exchange
economy, but one could similarly illustrate this in an economy with production.10
10 This approach to illustrating the “missing market” aspect of externalities was introduced by Kenneth Arrow
(1921-) (whom we previously mentioned in Chapter 16 as the 1972 Nobel Laureate who co-founded modern general
equilibrium theory) in “The Organization of Economic Activity: Issues Pertinent to the Choice of Market versus
Non-Market Allocations,” in Public Expenditure and Policy Analysis (Havenman, R. and J. Margolis, eds.), Chicago:
Markham (1970). A subsequent literature that we allude to in the Appendix to this chapter points out a technical
problem in this way of modeling externality markets, a problem we will for now glance over.
786
21B.3.1
Chapter 21. Externalities in Competitive Markets
Introducing Consumption Externalities into an Exchange Economy
In Chapter 16, we defined an exchange economy as a set of consumers denoted n = 1, 2, ..., N ,
with each consumer characterized fully by her endowments of each of M different goods as well as
her tastes summarized by utility functions defined over M goods (denoted m = 1, 2, ..., M ). An
exchange economy was then given simply by
11
n
M
{(en1 , en2 , ..., enm )}N
→ R1 } N
n=1 , {u : R
n=1 .
(21.18)
Because each consumer cares only about her own consumption of each of the goods (and because
there are no other agents like producers), there is no externality in this exchange economy. An
externality (in the absence of production) then arises when one consumer’s consumption directly
enters the utility function of another consumer. In principle, such consumption externalities in an
exchange economy could arise in every direction — with every consumer’s consumption of each
good entering every other consumer’s utility function.
We could then think of consumer n as consuming some of each of the M goods and being affected
by her “impression” of each other consumer’s consumption of each of the M goods. Suppose, for
instance, that we let xnij denote “person n’s impression of person j’s consumption of good i”. If
xnij enters person n’s utility function, then person j is generating a consumption externality when
consuming good i. But if each person’s consumption of each good potentially enters each person’s
utility function, then each person is in essence consuming N M different goods rather than M goods
as before. For instance, if N = 2 and M = 2, consumer 1 consumes (x111 , x121 , x112 , x122 ).
Exercise 21B.14 Which two of these four goods represent the consumption levels x11 and x12 that exist for
person 1 in an exchange economy without externalities?
We have therefore taken an economy with M goods and defined, for each person, N M goods
that enter her utility function. The exchange economy defined in equation (21.18) can then be
rewritten with consumption externalities as
21B.3.2
n
NM
{(en1 , en2 , ..., enm )}N
→ R1 } N
n=1 , {u : R
n=1 .
(21.19)
The Missing Markets in an Exchange Economy with Externalities
We have now introduced “impressions of other individuals’ consumption” explicitly as new goods.
But this implies that we have implicitly introduced production into the exchange economy —
because each time a consumer makes a decision to consume some of the M goods, she is “producing”
(N − 1) of these newly defined goods. When I consume good 1, I am producing an impression of
my consumption of good 1 that now potentially enters everyone else’s utility function. But our
exchange economy has no markets that set prices for such goods and thus no market mechanism to
govern my production decisions!
Suppose, for instance, that person j’s consumption of good i enters individual n’s utility function
in a positive way. In this case, person j is a producer of an output xnij , an output that consumers
like n would be willing to pay for but don’t since there is no market and no price. Alternatively,
suppose that xnij enters person n’s utility function negatively — implying consumer j emits a
negative consumption externality by consuming good i. In this case, we can view consumer j as
11 If you are uncomfortable with this notation, please review the discussion surrounding expression (16.1) in Chapter 16.
21B. The Mathematics of Externalities
787
using xnij as an input into the production of her own consumption of good i. But, once again because
there is no market for this input and thus no price, consumer j does not need to purchase the input
xnij when deciding how much of the good i to consume.
Exercise 21B.15 If there are two consumers and two goods, how many missing markets are there potentially? More generally, how many missing markets could there be when there are M goods and N consumers?
In some cases, externalities will take a form where the externality affects every consumer whereas
in other cases the externality may affect only some consumers. Suppose, for instance, that consumer
j is choosing good i that represents the number of car rides she takes — and each car ride emits
pollution that contributes to global warming. In that case, her car rides enter each consumer’s
utility function in the same quantity (even though different consumers will feel differently about
how bad this externality is). Put differently, in this example
xnij = xij for all n 6= j;
(21.20)
i.e. each individual other than j experiences the impact of j’s car rides in the same quantity. In
other cases, an externality is more “local” — affecting some individuals differently than others. For
instance, if j chooses good i that represents music played in the backyard, her immediate neighbors
are affected more than more distant neighbors. In this case, xnij will differ depending on the distance
between individual j and n.
21B.3.3
Introducing Property Rights and New Markets
In order establish the new markets that can price the externality effects within this exchange
economy, we have to begin by specifying a set of new property rights. If my car rides cause
pollution, do I have the right to pollute or do others have the right not to have pollution inflicted
on them? If I play loud music on my patio, do I have the right to do as much of this as I want to,
or do others have the right to not be bothered by my music? For efficiency purposes, however, it
turns out that what matters most is that property rights be established so that markets can form.
For now, we will illustrate in a simple example how markets price externalities when markets are
established, and we will return to a discussion of the extent to which it matters how property rights
are assigned in Chapter 27.
One way to think of how property rights are established in the new markets is to extend the
endowments for individuals to include endowments of the new goods. In this way, rights could
be distributed in a variety of ways — although we will typically think of rights being established
strictly one way or another; i.e. either someone has the right to pollute or the polluters have the
right not to be bothered by pollution unless they sell their rights. But once we have established
a system of property rights, we have arrived at an exchange economy that simply has more goods
than before. And none of the goods now appears in more than one utility function — which means
there is technically no more externality in the economy with the expanded set of markets. Since
we know that exchange economies without externalities are such that competitive equilibria are
efficient regardless of how many goods and consumers there are in the economy, the establishment
of these new markets therefore leads to an economy in which competitive equilibria are efficient,
with prices of the newly defined goods causing the emitters of externalities to take full account of
the marginal (social) costs and benefits of their actions.
788
Chapter 21. Externalities in Competitive Markets
21B.3.4
A Numerical Example
In Chapter 16, we worked through an example of a 2-person, 2-good exchange economy in which
1/4 3/4
3/4 1/4
(e11 , e12 ) = (3, 6), (e21 , e22 ) = (10, 4), u1 (x1 , x2 ) = x1 x2 and u2 (x1 , x2 ) = x1 x2 . Given that
only each individual’s own consumption appears in her utility function, this represents an exchange
economy without externalities. Suppose now, however, that consumption of good 1 by individual 1
enters individual 2’s utility function. Using our notation above, this implies that the good x211 —
individual 2’s perception of individual 1’s consumption of good 1, enters u2 . To keep our notation
in this example as simple as possible, let’s define x3 = x211 , and let individual 2’s utility function
be re-defined as
1/4 3/4
u2 (x1 , x2 , x3 ) = x1 x2 xγ3 .
(21.21)
Depending on whether γ is greater or less than zero, individual 1 is therefore now imposing a
positive or negative consumption externality on individual 2. When γ = 0, the example reduces to
our example from Chapter 16 with no externality.
We can first ask what the competitive equilibrium of this exchange economy will be. In the
absence of a market for x3 , however, nothing fundamental has changed from the way we calculated
the equilibrium of this economy in Chapter 16: Individual 1 will maximize the same utility function
subject to the same budget constraint as before and will thus have the same demand equations. Individual 2 will maximize the new utility function in equation (21.21) subject to the same constraints
as before, but xγ3 will simply cancel out as we solve for her demand equations — resulting in the
same demands as in Chapter 16. With both individuals exhibiting the same demands, we get the
same competitive equilibrium as before, with p2 /p1 = 3/2, (x11 , x12 ) = (9, 2) and (x21 , x22 ) = (4, 8).
Exercise 21B.16 Verify that individual 2’s demand functions for x1 and x2 are unchanged as a result of
the inclusion of x3 in her utility function.
Exercise 21B.17 Do you think the conclusion (in exercise 21B.16) that demands for x1 and x2 do not
change will hold regardless of what form the utility function takes?
But now suppose that a market is introduced for the good x3 (with price p3 ). Let’s begin by
thinking of the externality as negative (i.e. γ<0) — and suppose that property rights are assigned
such that individual 2 has the right to not experience the externality unless she agrees voluntarily to
do so. This implies that individual 1 will have to pay not only p1 for each unit of x1 she consumes —
but also p3 (since x11 = x3 ). The optimization problem for consumer 1 then becomes
(1−α)
max u1 (x1 , x2 ) = xα
1 x2
x1 ,x2
subject to p1 e11 + p2 e12 = (p1 + p3 )x1 + p2 x2 .
(21.22)
Solving this in the usual way, we get
x11 =
(1 − α)(p1 e11 + p2 e12 )
α(p1 e11 + p2 e12 )
and x12 =
.
p1 + p3
p2
(21.23)
Individual 2, on the other hand, will receive p3 for every unit of x3 that individual 1 emits, but,
since individual 2 is given the “property rights” to x3 , individual 2 chooses how much of x3 to sell.
The optimization problem for individual 2 then becomes
(1−β) γ
x3
max u2 (x1 , x2 , x3 ) = xβ1 x2
x1 ,x2 ,x3
subject to p1 e21 + p2 e22 + p3 x3 = p1 x1 + p2 x2 .
(21.24)
21B. The Mathematics of Externalities
789
Solving this, we get
x21 =
(1 − β)(p1 e21 + p2 e22 )
−γ(p1 e21 + p2 e22 )
β(p1 e21 + p2 e22 )
, x22 =
and x3 =
.
(1 + γ)p1
(1 + γ)p2
(1 + γ)p3
(21.25)
Exercise 21B.18 Verify these demand functions. (Hint: It becomes significantly easier algebraically to
first take logs of the utility function.)
Exercise 21B.19 Do the demand functions converge to those we derived in the absence of an externality
as the externality approaches zero (i.e. as γ approaches zero?
We can now solve for equilibrium prices. As in Chapter 16, we will be able to solve only for
relative prices and can therefore set one of the prices to 1. Suppose, then, we set
p1 = 1.
Setting demand equal to supply in the market for good 2 — i.e. setting
can then solve for p2 as
p2 =
(1 − α)(1 + γ)e11 + (1 − β)e21
.
α(1 + γ)e12 + (β + γ)e22
(21.26)
x12
+
x22
=
e12
+ e22 , we
(21.27)
In addition, it must be true that demand is equal to supply in the x3 market — where the
amount of x3 consumer 2 is willing to sell must be equal to the amount of x1 that consumer 1 wants
to consume — i.e. x11 = x3 . Solving this, we can get p3 in terms of p2 (with p1 again set to 1) —
p3 =
−γ(e21 + p2 e22 )
.
α(1 + γ)(e11 + p2 e12 ) + γ(e21 + p2 e22 )
(21.28)
In Table 21.2, we then calculate the competitive equilibrium prices and quantities when the
market for good x3 has been established. The table begins with negative values for γ — i.e. with
the case where individual 1’s consumption of good 1 imposes a negative externality on individual
2. As you move down the table, the externality becomes less severe, with no externality when
γ = 0. Finally, the table moves into positive values for γ — implying a positive externality on
individual 2 from the consumption of good 1 by individual 1. Notice that p3 is positive whenever
the consumption externality is negative — implying that the presence of a negative externality
results in individual 2 receiving compensation for suffering the negative effects of individual 1’s
consumption. But when the externality becomes positive, p3 becomes negative — implying that
now individual 2 compensates individual 1 for the positive effect x11 has on individual 2. Thus,
the establishment of the missing market results in individual 2 imposing a “tax” on individual 1’s
consumption of good 1 when the externality is negative and a “subsidy” when the externality is
positive.
Of course, just as in Chapter 16, it would not be reasonable to expect market prices to govern
exchange — either in the presence or in the absence of externalities — when there is literally only
one individual on each side of the market. The 2-person exchange economy simply provides a useful
tool with which to illustrate how markets set prices in general equilibrium. But the analysis above
continues to hold exactly the same way if we assume that there are many “type 1” and many
“type 2” individuals when competitive price taking behavior becomes more realistic. And for the
“two-person case” we have the Coase Theorem to fall back on – a theorem already mentioned in
Section A, and one we now examine a bit more closely.
790
Chapter 21. Externalities in Competitive Markets
γ
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Equilibrium with “Missing Market” Established
p1
p2
p3
x11
x12
x21
$1.00 $3.79
$6.64
2.52 1.70 10.48
$1.00 $2.72
$1.61
5.54 1.76
7.46
$1.00 $2.13
$0.64
7.21 1.85
5.79
$1.00 $1.76
$0.23
8.27 1.93
4.73
$1.00 $1.50
$0.00
9.00 2.00
4.00
$1.00 $1.31 -$0.15
9.54 2.07
3.46
$1.00 $1.17 -$0.25
9.94 2.14
3.06
$1.00 $1.05 -$0.32 10.27 2.21
2.73
$1.00 $0.96 -$0.38 10.53 2.28
2.47
x22
8.30
8.22
8.15
8.07
8.00
7.93
7.86
7.79
7.72
Table 21.2: α = 3/4, β = 1/4, (e11 , e12 ) = (3, 6), (e21 , e22 ) = (10, 4)
21B.4
Small “Markets” and the Coase Theorem
In Section A, we introduced the insight of Ronald Coase with respect to the types of externalities
that make us mad enough to take each other to court. We gave the example of me building an
addition to my house and you taking me to court because my addition would cast a shadow on
your beautiful swimming pool. It is precisely in such “small” settings that, even if we established
“markets” of the types we have discussed, there would not be much of a “market” since only one
or a few people would be operating on each side of the market. And, while we can theoretically
investigate what market prices would look like if they in fact arose, it is more realistic to think of
“bargaining” as the way in which externality issues would be resolved in such “markets”.
21B.4.1
Bargaining under Complete and Incomplete Information
Bargaining by definition does not happen in competitive settings – since in competitive settings
each consumer and producer is a price taker. We are therefore jumping a bit ahead of ourselves
as we think about bargaining under the Coase Theorem. You and I are decidedly not price takers
during our coffee as we discuss the level of compensation that you have to offer me to stop building
(if the judge ruled in my favor) or the level of compensation I will pay you to let me build (if the
judge ruled in your favor). Put differently, we are jumping ahead because we are thinking of a
“strategic” setting, one in which you and I have some real control over our economic environment.
Economists (and particularly game theorists) have, over the past few decades, arrived at a
well-defined theory of bargaining, some of which was directly inspired by Coase’s confidence that
bargaining in an atmosphere in which property rights have been fully clarified will lead to efficient
outcomes when externalities are involved. Some of that theory (just as some of the development of
game theory in Chapter 24) assumes that you and I have perfect information about each other’s
costs and benefits of my addition to my house. And under such circumstances, Coase appears to be
on solid ground: The theory predicts that you and I will in fact reach a bargain that will lead to the
efficient outcome under the conditions envisioned by Coase.12 Intuitively, this is not hard to see – if
12 In cases where income (or endowment) effects are important (as when tastes are not quasiliinear), we have to
be slightly more careful because “the” efficient outcome may differ depending on how property rights are assigned.
This is explored further in exercise 21.4.
21B. The Mathematics of Externalities
791
the story I told in Section A about how we will bargain our way to efficiency made sense, you have
the basic intuition. We can demonstrate this more formally once we have developed some game
theory tools – and illustrate how two individuals arrive at bargains under complete information in
end-of-chapter exercise 24.9.
21B.4.2
Bargaining under Incomplete Information
In Section A, however, we implicitly assumed what we have just made explicit: that you and I both
know what the costs to you are (relative to you solving your shaded pool problem in other ways) of
me adding to my house in a way that casts a shadow on your swimming pool and what the benefits
(relative to other ways of solving my need for additional housing) are to me of building the addition
in this way. Let’s denote your costs as c and my benefit as b. Efficiency dictates that I go ahead
with my addition if b > c, and we argued that, so long as property rights have been specified and
transactions costs are low, the efficient outcome will happen.
But suppose that you are not sure what b is and I am not sure what c is. Rather, you have beliefs
about b and I have beliefs about c. Let my beliefs be represented by 0 ≤ ρ(c) ≤ 1 for any c > 0,
with ρ(c) equal to the probability I place on your costs being less than or equal to c. Similarly, let
your beliefs be represented by 0 ≤ δ(b) ≤ 1 for any b > 0, with δ(b) equal to the probability that
you place on my benefits being less than or equal to b. Now suppose the judge rules in your favor –
i.e. you now have the right to a shadow-free pool and I cannot build my addition unless you agree
to it.
I will therefore come to coffee and offer you compensation based on my beliefs of what your
costs are. To arrive at an offer I make to you, I will have to calculate the offer p that maximizes
my expected payoff. My expected payoff from any offer p is the probability that the offer will be
accepted times the benefit I receive from having my offer accepted. For any offer p, I believe that
the probability that your true costs are less than or equal to p is ρ(p) – which implies that I believe
that the probability of you accepting my offer is ρ(p). The benefit I receive if the offer is accepted
is my benefit b from having the addition built minus the payment p I have to make to you – i.e.
the benefit I receive if the offer is accepted is (b − p). I therefore solve the following optimization
problem as I calculate my optimal offer given the beliefs I have:
max ρ(p)(b − p).
p
(21.29)
I will obviously not make an offer p > b, and you will not accept an offer p < c. But, depending
on what my beliefs are, I may well make an offer p∗ that maximizes my expected payoff but where
p∗ < c even though b > c. Thus, depending on my beliefs about your true underlying costs, the
addition may not get built if the judge rules in your favor despite the fact that building the addition
is efficient.13
Exercise 21B.20 Suppose the judge rules in my favor instead. What optimization problem do you solve
as you come over to have coffee in order to offer me a payment for not building the addition? Can it again
be the case that the efficient outcome does not happen for certain beliefs δ you might have about my true
benefit from the addition?
13 In Section B of Chapter 24, you will learn more about the equilibrium concept that we have implicitly just
applied, which we will call a Bayesian Nash Equilibrium. An example of this is also presented in end-of-chapter
exercise 21.2.
792
Chapter 21. Externalities in Competitive Markets
Depending on how we define what we mean by “transactions costs”, we now may or may not have
to amend the Coase Theorem. As stated in Section A, the theorem says that so long as property
rights are sufficiently specified in the presence of externalities, the efficient outcome will occur from
decentralized decisions if transactions costs are sufficiently low. As we have just seen, strategic
bargaining between individuals who understand the assignment of property rights in the presence
of externalities may not result in efficiency even when there are no transactions costs keeping the
individuals from getting together and bargaining. But the cost of obtaining information about the
relative costs and benefits from the externality may in itself be considered a transactions cost, in
which case we can leave the Coase Theorem as stated before.
Conclusion
This chapter is the first to have introduced an economic force that causes our First Welfare Theorem to break down: Markets, by themselves, cease to be efficient maximizers of social surplus
in the presence of externalities. While previous chapters may have given the impression that microeconomists see all forms of government intervention as inherently inefficient, we have now seen
that markets cannot operate efficiently in isolation. First, the very existence of markets presumes
an underlying system of property rights that, in practice, has almost always required the explicit
involvement of government. In the absence of such property rights, we are faced with what we have
called the “Tragedy of the Commons” when individual incentives lead to over-use of resources. And
where externalities arise, it is precisely because of the “Tragedy of the Commons”: We all “own”
the air (or, alternatively, none of us own it) – and as a result no one makes sure we pay for the
pollution we cause. We all “own” the roads, and so no one is charging us for the congestion we
contribute to when we drive during peak hours. The market failure that arises from externalities
is therefore caused by the “failure of a market to exist”.
This is not to say that markets themselves can always solve externality problems. Air pollution
is a problem because no market for air exists, but it is not exactly easy to establish such markets.
But policies aimed at correcting inefficiencies from externalities must ultimately do what markets
would do if they could be established fully: They must cause individual actors in the economy to face
the full marginal costs (and benefits) of their actions. We saw that this could in principle be done
through Pigouvian taxes and subsidies which force individuals to confront the larger social costs
and benefits of their private choices. We saw it could be done through the creative establishment of
markets like those for pollution vouchers which, this time through the need to purchase a voucher
if one intends to pollute, again forces polluters to pay for at least some of the social cost of their
production choices. Or it could be done through such policies as electronic tolls on roads. Or, as
Coase tells us, it could be done in the case of smaller externalities simply by clarifying through
property rights cases who in fact has a right to do what – and then rely on interested individuals
to bargain their way to efficiency. The key in all these policies, however, is to bring private and
social marginal costs in line with one another. Government policies (such as Pigouvian taxes and
subsidies), fostering of new markets (such as pollution vouchers) and clarifications of property rights
in the civil society (that can have individuals bargain outside the price-based market system) can
thus all contribute to greater efficiency in the presence of externalities.
In the upcoming chapter, we will see another important instance when competitive markets
by themselves will not result in efficient outcomes: the wide-spread case where information is not
shared uniformly by market participants. We will see that such asymmetric information results
in a new form of externality that can prevent important markets from forming — and that offer
21B. The Mathematics of Externalities
793
opportunities for non-market institutions to enhance efficiency.
Appendix: Fundamental Non-Convexities in the Presence of
Externalities
In our treatment of how the establishment of missing markets can restore efficiency in the presence
of externalities, we glanced over a technical problem that has become known as the problem of fundamental non-convexities. The essence of the problem is this: Suppose we reconsider our numerical
example of an exchange economy with a negative consumption externality from consumer 1’s consumption of good 1 (as we did in the chapter). Suppose further that we take the assumption that
consumer 2 has a right to not experience the externality and must be persuaded to sell that right by
accepting payment in proportion to the externality that is emitted. We know that if the price p3 is
zero, consumer 2 will not sell any rights to consume good 1 to consumer 1 (since consumer 2 would
then experience a negative externality without compensation). Now suppose that p3 >0 (as in the
equilibria we described in Table 21.2). What is to keep consumer 2 from wanting to sell an infinite
number of rights to pollute — thus making an infinite income to spend on consumption of goods
1 and 2? Put differently, if there is no limit on the number of rights that individual 2 can sell, a
positive price will cause the consumer to want to sell an infinite quantity of x3 while a non-positive
price will cause her to want to sell zero. No matter what p3 is set at, consumer 2 therefore prefers
a corner solution.14
But if consumer 2 will sell only zero or an infinite amount of x3 , no equilibrium in the x3
market exists — and the establishment of the x3 market with all rights assigned to the victim
of the negative externality does not in fact lead to a competitive equilibrium that eliminates the
inefficiency from the externality. In order for the equilibria that we discuss in Table 21.2 to emerge,
there must therefore be some limit to the number of rights that consumer 2 can sell.
The solution to this fundamental non-convexity problem lies in finding ways of “bounding” the
property rights in externality markets such that, for instance, victims of pollution cannot in fact
sell large or infinite amounts of these rights when the price is positive. While this is not easily
done in the context of defining externality markets in the way that we have done in our exchange
economy example, we have already shown how this in fact can be done when “rights” are defined
along the lines of pollution vouchers. Here, a limited number of these rights are allocated in the
economy, thus eliminating the problem of fundamental non-convexities.15
Exercise 21B.21 Why did our mathematical methods of solving for consumer 2’s demand for x3 not
uncover this problem?
End of Chapter Exercises
21.1 Consider the case of a positive consumption externality.
14 This is referred to as a “fundamental non-convexity” because it represents a non-convexity in the production
set for pollution rights. The problem of fundamental non-convexities in externality markets was first pointed out by
Starrett, D. (1972): “Fundamental Nonconvexities in the Theory of Externalities,” Journal of Economic Theory 4,
180-99.
15 This is explored in some detail by Boyd, J. and J. Conley (1997): “Fundamental Nonconvexities in Arrovian
Markets and a Coasian Solution to the Problem of Externalities,” Journal of Economic Theory 72, 388-407.
794
Chapter 21. Externalities in Competitive Markets
A: Suppose throughout this exercise that demand and supply curves are linear, that demand curves are equal
to marginal willingness to pay curves and that the additional social benefit from each consumption unit is k
and is constant as consumption increases.
(a) Draw two graphs with the same demand curve but one that has a fairly inelastic and one that has a fairly
elastic supply curve. In which case is the market output closer to the optimal output?
(b) Does the Pigouvian subsidy that would achieve the optimal output level differ across your two graphs in
part (a)?
(c) Draw two graphs with the same supply curve but one that has a fairly inelastic demand curve and one
that has a fairly elastic demand curve. In which case is the market output closer to the optimal output?
(d) Does the Pigouvian subsidy that would achieve the optimal output level differ across your two graphs in
part (c)?
(e) True or False: While the size of the Pigouvian subsidy does not vary as the slopes of demand and supply
curves change, the level of under-production increases as these curves become more elastic.
(f) In each of your graphs, indicate who benefits more from the Pigouvian subsidy – producers or consumers.
B: Suppose demand is given by xd = (A − p)/α and supply is given by xs = (B + p)/β.
(a) Derive the competitive equilibrium price and output level.
(b) Suppose that the marginal positive externality benefit is k per unit of output. What is the function for
the social marginal benefit SM B curve?
(c) What is the optimal output level?
(d) What is the Pigouvian subsidy? Show the impact it has on prices paid by consumers and prices received
by producers – and illustrate that it achieves the optimal outcome.
(e) Next, suppose that the total externality social benefit is given by SB = (δx)2 . Does the market outcome
change? What about the optimal outcome?
(f) Derive the Pigouvian subsidy now – and illustrate again that it achieves the social optimum.
21.2 The Coase Theorem is often applied in court cases where the parties seek to clarify who has the right to do
what in the presence of externalities. Consider again (as in the text discussion) the case of the addition to my house
that will then cast a shadow on your swimming pool. Suppose that my benefit from the addition is b, and the cost
you incur from my shadow is c. Suppose throughout this exercise that transactions costs are zero.
A: In this part of the exercise, suppose that you and I both know what b and c are.
(a) If we both know b and c, why don’t we just get together and try to settle the matter over coffee rather
than ending up in court?
(b) If the judge (who has to decide whether I have a right to build my addition) also knows b and c, propose
a sensible and efficient rule for him to use to adjudicate the case.
(c) Judges rarely have as much information as plaintiffs and defendants. It is therefore reasonable for the
judge to assume that he cannot easily ascertain b and c. Suppose he rules in my favor. What does Coase
predict will happen?
(d) What if he instead rules in your favor?
(e) In what sense will the outcome always be the same as it was in part (b) – and in what sense will it not?
B: Next, assume that I know b and you know c – but I do not know c and you do not know b.
(a) Suppose the judge rules in your favor, and I now attempt to convince you to let me build the addition
anyhow. I will come to your house and make an offer based on my belief that your cost is less than c with
probability ρ(c) = c/α. What offer will I make?
(b) For what combinations of b and c will the outcome be inefficient?
(c) Suppose instead that the judge ruled in my favor. You therefore come to my house to convince me not to
build the addition even though I now have the right to do so. You will make me an offer based on your
belief that my benefit from the addition is less than or equal to b with probability δ(b) = b/β. What offer
will you make?
(d) For what combinations of b and c will the outcome be inefficient?
(e) Explain how the cost of obtaining information might be considered a transactions cost – and the results
you derived here are therefore consistent with the Coase Theorem.
21B. The Mathematics of Externalities
795
21.3 We discussed in the text that the “market failure” that emerges in the presence of externalities can equally
well be viewed as a “failure of markets to exist”, and we discussed the related idea that establishing property rights
may allow individuals to resolve externality issues even when markets are not competitive.
A: We will explore this idea a bit further by asking whether there is a “right way” to establish property rights
in the case of pure consumption externalities.
(a) Suppose we consider the case where your consumption of music in your dorm room disturbs me next door.
Let x denote the number of minutes you choose to play music each day, and let e be the number of minutes
you are allowed to play music. If e is set at 0, who is given the “property rights” over the air on which
the soundwaves travel from your room to mine?
(b) What if e is set to 1,440 (which is equal to the number of minutes in a day)?
(c) The assignment of e in part (a) represents the extreme case where you have no right to play your music
while the assignment in (b) represents the polar opposite extreme where I have no right to peace and
quiet. Review the logic behind the Coase Theorem that suggests the efficient outcome will be reached
regardless of whether e = 0 or e = 1, 440 so long as transaction costs are low.
(d) Draw a graph with minutes of music per day on the horizontal axis – ranging from 0 to 1,440. Draw
a vertical axis at 0 minutes and another vertical axis at 1,440 minutes. Then illustrate your marginal
willingness to pay for minutes of music (measured on the left vertical axis) and my marginal willingness
to pay for reductions in the number of minutes of music (measured on the right axis) – and assume that
these are invariant to how e is set. What is the efficient number of minutes m∗ ?
(e) Since e = 0 and e = 1, 440 are two extreme assignments of property rights, we can now easily think of
many cases in between. Does the Coase Theorem apply also to these in between cases? Why or why not?
(f) From a pure efficiency standpoint, if the Coase Theorem is right, is there any case for any particular
assignment of e?
B: Suppose that your tastes can be described by the utility function u(x, y) = α ln x + y, where x is the number
of minutes per day of music and y is a composite consumption good. My tastes, on the other hand, can be
described by u(x, y) = β ln(1440 − x) + y, with (1440 − x) representing the number of minutes per day without
your music. Both of us have some daily income level I, and the price of y is 1 given that y is a composite good
denominated in dollars.
(a) Let e be the allocation of rights as defined in part A – i.e. e is the number of minutes that you are
permitted to play music without my permission. When x < e, I am paying you p(e − x) to play less than
you are allowed to – and when x > e, you are paying me p(x − e) for the minutes above your “rights”.
What is your budget constraint?
(b) What is my budget constraint?
(c) Set up your utility maximization problem using the budget constraint you derived in (a) – then solve for
your demand for x.
(d) Set up my utility maximization problem and derive my demand for x.
(e) Derive the p∗ we will agree to if transaction costs are zero – and derive the number of minutes of music
you will play. Does your answer depend on the level at which e was set?
(f) According to your results, how much music is played if I don’t care about peace and quiet (i.e. if β = 0?
How much is played if you don’t care about music – i.e. α = 0?
(g) True or False: The total number of minutes of music played does not depend on e – but you and I still
care how e is assigned.
* In exercise 21.3, we began to investigate different ways of assigning property rights in the presence of
externalities.
A: Consider again the case of you playing music that disturbs me.
(a) Begin with the assumptions in exercise 21.3 that led to the graph you drew in part (d) of that exercise.
Then suppose that the transaction cost of getting together is k. In your graph, indicate for what range of
e such a transaction cost will prohibit the efficient outcome from being reached?
21.4
(b) If e is assigned outside that range, what will be the outcome?
(c) Next, suppose income – or endowment – effects are important; i.e. tastes are not quasilinear. Did we
allow for that in exercise 21.3?
796
Chapter 21. Externalities in Competitive Markets
(d) Suppose in particular that such endowment effects matter for you but not for me – with music a normal
good for you. Illustrate in a graph what happens to the amount of music as e increases. What happens
to p?
(e) If endowment effects matter similarly for you and me, might it be the case that the agreed upon level of
music is once again unaffected by e?
(f) Is the Coase Theorem wrong in cases where endowment effects impact the amount of music that is played
as property rights are assigned differently?
(g) True or False: As long as transactions costs are zero, we will reach an efficient outcome – but that outcome
(i.e. the amount of music played) might differ depending on whether income effects are important.
B: Suppose first that our tastes are again those given in part B of exercise 21.3.
(a) If you have not done exercise 21.3, do so now and check whether the level of music played will depend on
the assignment of property rights e in the absence of transactions costs.
(b) Next, suppose that – instead of the tastes in exercise 21.3 – your tastes can be described by the utility
function u(x, y) = xα y (1−α) (where α lies between 0 and 1). My tastes remain unchanged. How much
music will be played? Does your answer depend on e – and does the equilibrium price p∗ depend on e?
(c) Next, suppose that my utility function is also Cobb-Douglas, taking the form u(x, y) = (1440 − x)β y (1−β) .
Derive again the amount of music that will be played (assuming zero transactions costs). Does your answer
depend on e? Does the equilibrium price depend on e?
(d) Explain your results intuitively.
(e) In section 21B.3.4, we went through a numerical exercise to illustrate how the establishment of property
rights in the presence of externalities will resolve the “market failure” in a simple exchange economy.
Review the example in the text prior to proceeding. Note that in the text we assigned the property rights
in the new market to person 2 – the victim of the externality. But we could have assigned property rights
in many other ways (as suggested in our music example). Define x3 once again as the impression of person
1’s consumption of x1 on person 2; i.e. x3 = x211 . We can establish a market for the good x3 by endowing
individual 1 with e3 units of x3 . This means that individual 1 can produce up to e3 units of x3 – which
is the same as saying that individual 1 can consume up to e3 units of x1 – without having to pay the
market price p3 . But if he wants to produce any more x3 , he must pay individual 2 p3 for each additional
unit above e3 . Similarly, under the endowment of e3 for individual 1, individual 2 must pay p3 per unit
to individual 1 for any amount of x3 that falls below e3 – and receives p3 for any amount of x3 above e3 .
In the numerical example of the text, what did we implicitly set e3 to?
(f) Write down individual 1’s budget constraint when he is assigned e3 in property rights. (Hint: If x1 < e1 ,
individual 1 will earn p3 (e3 − x1 ) but if x1 > e1 , he will have to pay p3 (x1 − e3 ) which is equivalent to
saying he will earn p3 (e3 − x1 ).)
(g) Next, write down individual 2’s budget constraint.
(h) If you substitute your answer to (e) into the budget constraints in (f) and (g), you should end up with
the budget constraints we used in the numerical example of the text. Do you?
(1−α)
(1−β)
. Suppose further that p1 = 0, p2 = 1
and u2 = (1440 − x3 )β x2
(i) Now suppose that u1 = xα
1 x2
and p3 = p, and that e12 = e22 . Can you now interpret the general equilibrium model as modeling our case
of you (person 1) bothering me (person 2) with music?
(j) Solve for p and x3 (which is equal to x11 ). Do you get the same answer as you got when you assumed
Cobb-Douglas tastes for both of us in part (c)?
21.5 Everyday Exercise: Children’s Toys and Gucci Products: In most of our development of consumer theory, we
have assumed that tastes are independent of what other people do. This is not true for some goods. For instance,
children are notorious for valuing toys more if their friends also have them – which implies their marginal willingness
to pay is higher the more prevalent the toys are in their peer group. Some of my snooty acquaintances, on the
other hand, like to be the center of attention and would like to consume goods that few others have. Their marginal
willingness to pay for these goods thus falls as more people in their peer group consume the same goods.16
A: The two examples we have cited are examples of positive and negative network externalities..
16 Such goods are examples of Veblen goods. We previously mentioned these in an exercise in Chapter 7 as goods
whose demand can slope up without being Giffen goods.
21B. The Mathematics of Externalities
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(a) Consider children’s toys first. Suppose that, for a given number N of peers, demand for some toy x is
linear and downward sloping – but that an increase in the “network” of children (i.e. an increase in
N ) causes an upward parallel shift of the demand curve. Illustrate two demand curves corresponding to
network size levels N1 < N2 .
(b) Suppose every child at most buys one of these toys which are produced at constant marginal cost. For
a combination of p and x to be an equilibrium, what must be true about x if the equilibrium lies on the
demand curve for network size N1 ?
(c) Suppose you start in such an equilibrium and the marginal cost (and thus the price) drops. Economists
distinguish between two types of effects: A direct effect that occurs along the demand curve for network size
N1 , and a bandwagon effect that results from increased demand due to increased network size. Label your
original equilibrium A, the “temporary” equilibrium before network externalities are taken into account
as B and your new equilibrium (that incorporates both effects) as C. Assume that this new equilibrium
lies on the demand curve that corresponds to network size N2 .
(d) How many toys are sold in equilibrium C? Connect A and C with a line labeled D. Is D the true demand
curve for this children’s toy? Explain.
(e) If you were a marketing manager with a limited budget for a children’s toy company, would you spend
your budget on aggressive advertising early as the product is rolled out or wait and spread it out? Explain.
(f) Now consider my snooty acquaintances who like Guuci products more if few of their friends have them. For
any given number of friends N that also have Gucci products, their demand curve is linear and downward
sloping – but the intercept of their demand curve falls as N increases. Illustrate three demand curves for
N1 < N 2 .
(g) Assume for convenience that everyone buys at most 1 Gucci product. Identify an initial equilibrium A
under which N1 Gucci products are sold at some initial price p – and then a second equilibrium C at which
N2 Gucci products are sold at price p′ < p. Can you again identify two effects – a direct effect analogous
to the one you identified in (c) and a snob effect analogous to the bandwagon effect you identified for
children’s toys? How does the snob effect differ from the bandwagon effect?
(h) True or False: Bandwagon effects make demand more price elastic while snob effects make demand less
price elastic.
(i) In exercise 7.9, we gave an example of an upward sloping demand curve for Gucci products, with the
upward slope emerging from the fact that utility was increasing in the price of Gucci products. Might the
demand that takes both the direct and snob effects into account also be upward sloping in the presence
of the kinds of network externalities modeled here?
B: Consider again the positive and negative network externalities described above.
(a) Consider first the case of a positive network externality such as the toy example. Suppose that, for a
given network size N , the demand curve is given by p = 25N 1/2 − x. Does this give rise to parallel linear
demand curves for different levels of N , with higher N implying higher demand?
(b) Assume that children buy at most one of this toy. Suppose we are currently in an equilibrium where
N = 400. What must the price of x be?
(c) Suppose the price drops to $24. Isolate the direct effect of the price change – i.e. if child perception of N
remained unchanged, what would happen to the consumption level of x?
(d) Can you verify that the real equilibrium (that includes the bandwagon effect) will result in x = N = 576
when price falls to $24? How big is the direct effect relative to the bandwagon effect in this case?17
(e) Consider next the negative network externality of the Gucci example. Suppose that, given a network of
size N , the market demand curve for Gucci products is p = (1000/N 1/2 )− x. Does this give rise to parallel
linear demand curves for different levels of N , with higher N implying lower demand?
(f) Assume again that no one buys more than one Gucci item. Suppose we are currently in equilibrium with
N = 25. What must the price be?
(g) Suppose the price drops to $65. Isolate the direct effect of the price change – i.e. if people’s perception of
N remained unchanged, what would happen to the consumption level of x?
(h) Can you verify that the real equilibrium (that includes the snob effect) will result in x = N = 62? How
big is the direct effect relative to the snob effect in this case?
17 For a more detailed analysis of the quite interesting demand curve that arises under this network externality, see
a similar example in exercise 21.8.
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Chapter 21. Externalities in Competitive Markets
(i) Although the demand curves for a fixed level of N are linear, can you sketch the demand curve that
includes both direct and snob effects?
21.6 Business Application: Fishing in the Commons: In the text, we introduced the notion of the Tragedy of the
Commons – and found its source in the emergence of externalities when property rights are not well established.
This exercise demonstrates the same idea in a slightly different way.
A: Consider a self-contained lake which is home to fish that are sold on the market at p. Suppose the primary
input into fishing this lake is nets that are rented at a weekly rate of r, and suppose the single input production
frontier for fish has decreasing returns to scale.
(a) Draw a graph with fishing nets on the horizontal axis and fish on the vertical. Illustrate the marginal
product of fishing nets.
(b) Recalling the relationship between “marginal” and “average” quantities, add the average product curve
to your graph.
(c) If you own the lake, what is the relationship between the marginal product of fishing nets and prices (p, r)
assuming you maximize profit?
(d) Illustrate the profit maximizing quantity of nets n∗ on your graph. Then, on a graph below it that plots
the production frontier for fish, illustrate the number of fish x∗ that are brought to market.
(e) Suppose you instead charge a weekly fee for every fishing net that fishermen bring to your lake. Does the
number of fish produced and nets used change?
(f) Next consider a nearby lake that is identical in every way except that it is publicly owned – with no one
controlling who can come onto the lake to fish. Assuming all nets are used with the same intensity, each
fishing net that is brought onto the lake can then be expect to catch the average of the total weekly catch.
Illustrate on your graphs how many nets n will be brought onto this lake – and how many fish x this
implies will be brought to market each week.
(g) Which lake yields more fish per week? Which lake is being harvested for fish efficiently?
(h) Suppose that what matters is not just the current crop of fish but also its implication for the future
fish population of the lake. Explain how the privately owned lake is likely to house a relatively constant
population of fish over time while the publicly owned lake is likely to run out of fish as time passes.
(i) The trade in elephant trunks – or ivory – has decimated much of the elephant population in some parts
of Africa but not in others, with hunters often slaughtering entire herds, removing the trunks and leaving
the rest. In some parts of Africa, the land on which elephants roam is public property – in other parts it
is privately owned with owners allowed to restrict access. Can you guess from our lake example what is
different about the parts of Africa where elephant herds are stable compared to those parts where they
are nearing extinction?
(j) Why do you think that wild Buffalo in the American West are nearly extinct while domesticated cattle
are plentiful in the same region?
B: Let n again denote the fishing nets used in the lake and assume that r is the weekly rental cost per net. The
number of fish brought out of the lake per week is x = f (n) = Anα where A > 0 and 0 < α < 1, and fish sell
on the market for p.
(a) Suppose you own the lake and you don’t let anyone other than yourself fish. How many fish will you pull
out each week assuming you maximize profit?
(b) Suppose instead you allow others to fish for a fee per net – and you want to maximize your fees. Will
more or fewer fish be pulled out each week?
(c) Next, consider the identical lake that has just been discovered near yours. This lake is publicly owned,
and anyone who wishes to can fish there. How many fish per week will be pulled out from that lake?
(d) Suppose A = 100, α = 0.5, p = 10 and r = 20. How many fish are harvested per week in (a), (b) and (c)?
How many nets are used in each case?
(e) What is the weekly rental value of the lake? If we count all your costs – including the opportunity cost of
owning the lake, how much weekly profit do you make if you are the only one to fish on your lake?
(f) How much profit (including the opportunity cost of fishing on the lake yourself) do you make if you allow
others to fish on your lake for a per-net-fee? How much profit do the fishermen who pay the fee to fish on
your lake make?
(g) How much profit do the fishermen who fish on the publicly owned nearby lake make?
21B. The Mathematics of Externalities
799
(h) If the government auctioned off the nearby lake, what price do you think it would fetch if the weekly
interest rate is 0.12% or 0.0012?
(i) If the government auctioned off the nearby lake with the condition that the same number of fish per week
need to be brought to market as before, what price would the lake fetch?
21.7 Business and Policy Application: The Externality when Fishing in the Commons: In exercise 21.6, we showed
that free access to a fishing lake causes overfishing because fishermen will continue to fish until the cost of inputs
(i.e. fishing nets, in our example) equals average rather than marginal revenue product.
A: Suppose that the lake in exercise 21.6 is publicly owned.
(a) What is the externality that fishermen impose on one another in this lake?
(b) Seeing the problem as one involving this externality, how would you go about setting a Pigouvian tax on
fishing nets to remedy the problem? What information would you have to have to calculate this?
(c) Suppose instead that the lake is auctioned off to someone who then charges per-net fees to fishermen who
would like to fish on the lake (as in A(e) of exercise 21.6). How do you think the fees charged by a profit
maximizing lake-owner compare to the optimal Pigouvian tax?
(d) Do you think it is easier for the government to collect the information necessary to impose a Pigouvian
tax in part (b) or for a lake-owner to collect the information necessary to impose the per-net fees in part
(c). Who has the stronger incentive to get the correct information?
(e) How would the price of the lake that the government collects in (c) compare to the tax revenues it raises
in (b)?
(f) Suppose instead that the government tries to solve the externality problem by simply setting a limit on
per-net fishing licenses that fishermen are now required to use when fishing on the public lake. If the
government sets the optimal cap on licenses and auctions these off, what will be the price per license?
(g) What do each of the above solutions to the Tragedy of the Commons share in common?
(h) Legislators who represent political districts (such as Congressmen in the U.S. House of Representatives)
can be modeled as competing for pork barrel projects to be paid for by the government budget. Could
you draw an analogy between this and the problem faced by fishermen competing for fish in a public lake?
(This is explored in more detail in end-of-chapter exercise 28.2 in Chapter 28.)
B: * Let N denote the total number of fishing nets used by everyone and X = f (N ) = AN α the total catch
per week. As in exercise 21.6, let r be the weekly rental cost per net, let p be the market price for fish and let
A > 0 and 0 < α < 1.
(a) The lake is freely accessible to anyone who wants to fish. How much revenue does each individual fisherman
make when he uses one net?
(b) What is the loss in revenue for everyone else who is fishing the lake when one fisherman uses one more
net?
(c) Suppose that each fisherman took the loss of revenue to others into account in his own profit maximization
problem when choosing how many nets n to bring. Write down this optimization problem. Would this
solve the externality problem?
(d) A Pigouvian tax is optimally set to be equal to the marginal social damage an action causes when evaluated
at the optimal market level of that action. Evaluate your answer to (b) at the optimal level of N to derive
the optimal Pigouvian tax on nets.
(e) Suppose that all fishermen just consider their own profit but that the government has imposed the Pigouvian per-net tax you derived in (d). Write down the fisherman’s optimization problem and illustrate its
implications for the overall level of N . Does the Pigouvian tax achieve the efficient outcome?
(f) Suppose the government privatized the lake and allowed the owners to charge per-net fees. The owner
might do the following: First, calculate the maximum profit (not counting the rental value of the lake)
he would be able to make by simply fishing the lake himself with the optimal number of nets – then set
the fee per net at this profit divided by the number of nets he himself would have used. What per-net fee
does this imply?
(g) Compare your answer to (f) to your answer to (d). Can you explain why the two are the same?
(h) Suppose A = 100, α = 0.5, p = 10 and r = 20. What is the optimal Pigouvian (per-net) tax and the
profit maximizing per-net fee that an owner of the lake would charge?
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Chapter 21. Externalities in Competitive Markets
21.8 Business Application: Network Externalities and the Battle between Microsoft and Apple: Many markets
related to technology products operate in the presence of network externalities because the value of such products
to consumers depends on how many other consumers are in the “network” of consumers. For instance, an internet
connection would not be nearly as useful if no one else in the world was connected to the internet; a telephone
becomes more useful the more other people also have telephones; and a computer operating system becomes more
useful the more others use it – because then the market for software that runs on this operating system increases
which in turn fosters greater software innovation for that platform. Assume throughout that we are analyzing the
consumer market for computers and that a consumer buys at most one computer.
A: Consider the market for PC’s when the Microsoft Windows system first competed with the Apple Macintosh
platform in the 1980’s. Microsoft and Apple pursued very different strategies: Microsoft licensed the Windows
platform to lots of PC makers who competed with one another and thus drove down the price of PCs. Apple, on
the other hand, did not license its Macintosh operating system – and sold it only with its own Apple computers
that were more expensive.
(a) Suppose that people vary greatly in their interest for buying a personal computer, but their willingness
to pay for a computer increases with the square root of the size of the “network” of others who use a
computer with the same operating system; i.e. if someone’s willingness to pay for a computer is B when
no one else is in the “network”, her willingness to pay for the same computer is BN 1/2 when the network
has N people. Pick three different levels of N – with N1 < N2 < N3 – and illustrate the linear aggregate
demand curves – D1 , D2 and D3 – that correspond to these levels of N for a computer with a particular
operating system.
(b) Suppose the demand curve D1 tells us that N1 computers are demanded at price p. In what sense is this an
equilibrium in which consumers are taking into account the network externality in their decision-making?
(c) Now suppose the price drops from p to p′ . If everyone assumes that the network size remains fixed at N1 ,
illustrate how many more computers will be sold. Why can this not be an equilibrium in the same way
that our previous situation was an equilibrium?
(d) Now take into account that people will realize that the network is growing as price falls. What will happen
if the number of computers demanded at p′ on D3 is N3 ? Illustrate the new equilibrium – and explain
why some economists say that network externalities give rise to a bandwagon effect in addition to a direct
price effect.
(e) How do you think the process of moving from our initial equilibrium to the final equilibrium unfolds over
time as price falls from p to p′ ? True or False: Network externality of this kind cause demand to become
more price elastic.
(f) Microsoft got a head start with its licensing policy that created competition and thus sharply falling prices
in the PC market – while Apple’s computers were perpetually priced above PC’s. Can you use this model
to explain how Microsoft’s Windows operating system became the dominant operating system?
(g) Suppose that the quality of Apple computers is now far better than any competing PC’s – and that it can
be priced competitively. Why is this not enough for Apple to gain dominance in the computer market?
How might you argue that the network externality you analyzed has led to an inefficient market outcome?
(h) Explain the following statement made by a technology company executive: “In the quickly moving tech
market, it is usually better to be first rather than best.”
(i) In a recent update to its operating system, Apple introduced a new feature that allows users to switch
between the traditional Macintosh operating system and the Microsoft Windows operating system. Do
you think this was a good move in light of what this exercise has told us about network externalities?
B: Now consider the type of network externality described in part A more carefully. Suppose that the aggregate
demand function for computers is given by x = (AN 1/2 − p)/α.
(a) Does this demand function give rise to the parallel demand curves (for different levels of network size) you
analyzed in part A?
(b) The consumer side of the market is in equilibrium if the network size N is equal to the number of computers
sold. Use this to derive the actual demand curve P (x) that takes the network externality fully into account.
(c) Suppose A > 2α. What is the shape of this demand curve? Explain.
(d) Check your answer to (c) by graphing the demand function when A = 100 and α = 1. Continue with
these parameter values for the rest of the exercise.
21B. The Mathematics of Externalities
801
(e) In models like this, we say that an equilibrium is stable if it does not lie on an upward sloping portion
of the demand curve. Can you guess why? (Hint: Suppose that x∗ is the equilibrium quantity on the
upward sloping part of demand for some price p∗ . Imagine what would happen if slightly more than x∗
were bought, and what would happen if slightly less than x∗ would be bought.)
(f) Suppose the supply curve is horizontal at p = 2, 000. Our model implies there are three equilibria – 2 that
are stable and one that is not stable. What network sizes are associated with each of these equilibria?
(g) Suppose that we begin in the equilibrium in which no one owns a computer and the marginal cost of
producing computers is $2,000. Why might firms launch an aggressive campaign in which they give away
computers before selling them in stores? How many might they give away to “jump-start” the market?
21.9 Business Application: Pollution that increases firm costs – the Market Outcome: In the text, we assumed for
convenience that the ill effects of pollution are felt by people other than producers and consumers. Consider instead
the following case: An entire competitive industry is located around a single lake that contains some vital property
needed for the production of x. Each unit of output x that is produced results in pollution that goes into the lake.
The only effect of the pollution is that it introduces a chemical into the lake – a chemical that requires firms to
re-inforce their pipes to keep them from corroding. The chemical is otherwise harmless to the population as well as
to all wildlife in the area.
A: We have now constructed an example in which the only impact of pollution is on the firms that are creating
the pollution. Suppose that each unit of x that is produced raises every firm’s fixed cost by δ.
(a) Suppose all firms have identical decreasing returns to scale production processes, with the only fixed cost
created by the pollution. For a given amount of industry production, what is the shape of an individual
firm’s average cost curve?
(b) In our discussion of long run competitive equilibria, we concluded in Chapter 14 that the long run industry
supply curve is horizontal when all firms have identical cost curves. Can you recall the reason for this?
(c) Now consider this example here. Why is the long run industry supply curve now upward sloping despite
the fact that all firms are identical?
(d) In side-by-side graphs of a firm’s cost curves and the (long run) industry supply and demand curves,
illustrate the firm and industry in long run equilibrium.
(e) Usually we can identify producer surplus – or firm profit – as an area in the demand and supply picture.
What is producer surplus here? Why is your answer different from the usual?
(f) In chapter 14, we briefly mentioned the term decreasing cost industries – industries in which the long run
industry supply curve is downward sloping despite the fact that all firms might have identical production
technologies. Suppose that in our example the pollution causes a decrease rather than an increase in
fixed costs for firms. Would such a positive externality be another way of giving rise to a decreasing cost
industry?
B: * Suppose that each firm’s (long run) cost curve is given by c(x) = βx2 + δX where x is the firm’s output
level and X is the output level of the whole industry. Note that x is contained in X – and thus we could write
the cost function as c(x) = βx2 + δx + δX where X is the output produced by all other firms. When each firm
is small relative to the industry, however, the impact of a single firm’s pollution output on its own production
cost is negligible – and it is a good approximation (that makes the problem a lot easier to solve) to simply write
a single firm’s cost curve as c(x) = βx2 + δX. Furthermore, if all firms are identical, it is reasonable to assume
that all firms produce the same output level x. Letting N denote the number of firms in the industry, we can
therefore write X = N x and re-write the cost function for an individual firm as c(x) = βx2 + δN x.
(a) How is our treatment of a producer’s contribution to her own costs similar to our “price-taking” assumption
for competitive firms?
(b) Derive the marginal and average cost functions for a single firm (using the final version of our approximate
cost function). (Be careful to realize that the second part of the cost function is, from the firm’s perspective,
simply a fixed cost.)
(c) Assuming the firm is in long run equilibrium, all firms will make zero profit. Use your answer to (b) to
derive the output level produced by each firm as a function of δ, β, N and x.
(d) Since all firms are identical, in equilibrium the single firm we are analyzing will produce the same as each
of the other firms – i.e. x = x. Use this to derive a single firm’s output level x(N ) as a function of δ,
N , and β. What does this imply about the equilibrium price p(N ) (as a function of δ and N ) given that
firms make zero profit in equilibrium?
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Chapter 21. Externalities in Competitive Markets
(e) Since each firm produces x(N ), multiply this by N to get the aggregate output level X(N ) – then invert
it to get the number of firms N (X) as a function of β, δ and X.
(f) Substitute N (X) into p(N ) to get a function p(X). Can you explain why this is the long run industry
supply curve with free entry and exit?
(g) Suppose the aggregate demand for X is given by the demand curve pD (X) = A/(X 0.5 ). Set the industry
supply curve equal to the demand curve to get the equilibrium market output X ∗ (as a function of A, δ
and β).18
(h) Use your answer to (g) to determine the equilibrium price level p∗ (as a function of A, δ and β).
(i) Use your answer to (g) to determine the equilibrium number of firms N ∗ (as a function of A, δ and β).
(j) Suppose that β = 1, δ = 0.01 and A = 10, 580. What are X ∗ , p∗ and N ∗ ? How much does each individual
firm produce? (Do exercise 21.10 to compare these to what is optimal.)
21.10 Policy Application: Pollution that increases firm costs – Barney’s Solution: Consider the same situation as
the one described in exercise 21.9.
A: Assume again that the only impact of pollution is that it increases firm fixed costs by δ for every unit of x
that is produced in the industry.
(a) Suppose there are N firms in the equilibrium you described in exercise 21.9. What is the pollution related
cost of firm i producing one more unit of x?
(b) How much of this pollution related cost does firm i not take into account? If firm i is one of a large
number of firms, is it a good approximation to say that firm i does not take any of the pollution related
cost into account? How is this similar to our “price-taking” assumption for competitive firms?
(c) Suppose that our benevolent social planner Barney can tell firms what to count as costs. Illustrate how
Barney’s suggestion for each firm’s marginal cost curve is related to the marginal cost curve firms would
otherwise use (given a fixed number N of firms in the industry)?
(d) What does your answer imply about the relationship between the firm’s AC curve and Barney’s suggestion
for what the firm’s AC curve should be?
(e) True or False: If firms used Barney’s suggested cost curves, the long run industry supply curve would be
upward sloping as you should have concluded in exercise 21.9 it is in the absence of Barney – but now it
would lie above where it was in exercise 21.9.
(f) True or False: Under the efficient outcome, the industry would produce less at a higher price.
(g) If a single corporation acquired all the firms around the lake, would that corporation take the costs of
pollution into account more like Barney or more like the individual competitive firms? (In parts of exercise
23.11, you’ll be asked to revisit this in the context of such a monopoly.)
B: * Consider the same set-up as in part B of exercise 21.9. In the previous case where we derived the market
equilibrium, we said that – in a model with many firms – it was reasonable to model each individual firm as
not taking its own impact of pollution into account and to simply model the cost function as c(x) = βx2 + δN x
(where the latter entered as a fixed cost).
(a) Now consider the cost function that benevolent Barney would use for each firm: From the social planner’s
perspective, the firm’s variable costs (captured by βx2 ) would still matter, as would the fixed cost from
pollution (captured by δN x where x is the amount produced by each firm and N is the number of firms
in the industry.) But Barney also cares about the following: each unit of x produced by firm i causes
an increase in costs of δ for each of the N firms – which implies that the pollution cost Barney would
consider firm i as imposing on society is δN x. This implies that Barney’s cost function for each firm is
cB (x) = βx2 + δN x + δN x. Derive from this the marginal and average cost functions that Barney would
use for each firm (being sure to not treat the last term as a fixed cost.)
(b) Repeat parts (c) through (i) from exercise 21.9 using the cost functions Barney would use for each firm
to arrive at N ∗ , p∗ and X ∗ .
(c) Compare your answers to those from exercise 21.9. How do they differ?
18 Note that the demand function is one that would emerge from utility maximization of the utility function
U (x, y) = 2Ax0.5 + y (where y is a composite good). Thus, it can be viewed as emerging from a representative agent
with tastes that are quasilinear in x – and thus represents a true aggregate marginal willingness to pay as well as an
uncompensated demand curve. See Chapter 15 for a review of this.
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803
(d) Suppose, as in part (j) of exercise 21.9 that β = 1, δ = 0.1 and A = 10, 580. What are X ∗ , p∗ and N ∗ ?
How much does each individual firm produce?
(e) Compare these to your answers in exercise 21.9. Can you give an intuitive explanation for why these
answers differ despite the fact that pollution only affects the firms in the indusry?
(f) What is the Pigouvian tax that is required in order for competitive firms to implement the equilibrium
you just calculated in (d)? What price does this imply consumers would pay and what price does it imply
producers would receive?
(g) Verify that your Pigouvian tax in fact results in prices for consumers and the industry that lead them to
demand and supply the output level you calculated in part (d). (Note: You will need to refer back to your
answers to exercise 21.9 to do this part.)
21.11 Policy Application: Pollution that increases firm costs – Policy Solutions: This exercise continues to build
on exercises 21.9 and 21.10. Assume the same basic setup of firms located around a lake producing pollution that
causes the fixed costs of all firms to increase.
A: Continue to assume that each output unit that is produced results in an increase of fixed costs of δ for all
firms in the industry.
(a) Begin by illustrating the market demand and long run industry supply curves, labeling the market equilibrium as A.
(b) Next, without drawing any additional curves, indicate the point B in your graph where the market would
be producing if firms were taking the full cost of the pollution they emit into account.
(c) Illustrate the Pigouvian tax that would be necessary to get the market to move to equilibrium B.
(d) Suppose N ∗ is the number of firms in the industry in the market outcome, N opt is the optimal number of
firms and δ continues to be as defined throughout. What does the government have to know in order to
implement this Pigouvian tax? Is what the government needs to know easily observable prior to the tax?
(e) Where in your graph does consumer surplus before and after the tax lie?
(f) Keeping in mind what you concluded in exercise 21.9, has (long run) producer surplus – or long run
industry profit – changed as a result of the tax?
(g) True or False: The pollution cost under the Pigouvian tax is, in this example, equal to the tax revenue
that is raised under the tax.
(h) Is there additional pollution damage under the market outcome (in the absence of the tax)?
(i) Is there a deadweight loss from not using the tax?
(j) Suppose the government instead wanted to impose a cap-and-trade system on this lake – with pollution
permits that allow a producer to produce the amount of pollution necessary to produce one unit of
output. What is the “cap” on pollution permits the government would want to impose to achieve the
efficient outcome? What would be the rental rate of such a permit when it is traded?
(k) What would the government have to know to set the optimal cap on the number of pollution permits?
B: Continue with the functional forms for costs and demand as given in exercises 21.9 and 21.10. Suppose, as
you did in parts of the previous exercises, that β = 1, δ = 0.1 and A = 10, 580 throughout this exercise.
(a) If you have not already done so in part (f) of exercise 21.9, determine the Pigouvian tax that would cause
producers to behave the way the social planner would wish for them to behave. What price will consumers
end up paying and what price will firms end up keeping under this tax?
(b)
**
Calculate (for our numerical example) consumer surplus with and without the Pigouvian tax. (Skip
this if you are not comfortable with integral calculus.) Why is (long run) producer surplus – or long run
profit in the industry – unchanged by the tax?
(c) Determine the total cost of pollution before and after the tax is imposed.
(d) Determine tax revenue from the Pigouvian tax.
(e) What is the total surplus before and after the tax – and how much deadweight loss does this imply in the
absence of the tax?
(f) Suppose next that the government instead creates a tradable pollution permits – or voucher – system in
which one voucher allows a firm to produce the amount of pollution that gets emitted from the production
of 1 unit of output. Derive the demand curve for such vouchers.
(g) What is the optimal level of vouchers for the government to sell – and what will be the rental rate of the
vouchers if the government does this?
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Chapter 21. Externalities in Competitive Markets
21.12 Policy Application: Social Norms and Private Actions: When asked to explain our actions, we sometimes
simply respond by saying “it was the right thing to do.” The concept of “the right thing to do” is one that is often
formed by observing others – and the more we see others “do the right thing,” the more we believe it is in fact “the
right thing to do”. In such cases, my action “to do the right thing” directly contributes to the social norm that
partially governs the behavior of others – and we therefore have an example of an externality.
A: Consider for instance the use of observably “green” technology – such as driving hybrid cars. Suppose there
are two types of car-buyers: (1) a small minority of “greenies” for whom green technology is attractive regardless
of what everyone else does – and whose demand for green cars is therefore independent of how many others are
using green cars; and (2) the large majority of “meanies” who don’t care that much about environmental issues
but do care about being perceived as “doing the right thing.”
(a) Draw a graph with the aggregate demand curve D0 for the “greenies.” Assume that green cars are
competitively supplied at a market price p∗ – and draw in a perfectly elastic supply curve for green cars
at that price.
(b) There are two types of externalities in this problem. The first arises from the positive impact that green
cars have on the environment. Suppose that the social marginal benefit associated with this externality is
some amount k per green car and illustrate in your graph the efficient number of cars x1 that this implies
for “greenies”. Then illustrate the Pigouvian subsidy s that would eliminate the market inefficiency.
(c) The second externality emerges in this case from the formation of social norms – a form of network
externality. Suppose that the more green cars the “meanies” see on the road, the more of them become
convinced that it is “the right thing to do” to buy green cars even if they are somewhat less convenient
right now. Suppose that the “meanies’s” linear demand D1 for green cars when x1 green cars are on
the road has vertical intercept below (p∗ − k). In a separate graph, illustrate D1 – and then illustrate a
demand curve D2 that corresponds to the demand for green cars by “meanies” when x2 (> x1 ) green cars
are on the road. Might D2 have an intercept above p∗ ?
(d) Does the subsidy in (b) have any impact on the behavior of the “meanies”? In the absence of the network
externality, is this efficient?
(e) How can raising the subsidy above the Pigouvian level have an impact far larger than one might initially
think from the imposition of the original Pigouvian tax? If the network externalities are sufficiently strong,
might one eventually be able to eliminate the subsidy altogether and see the majority of “meanies” use
green cars anyhow?
(f) Explain how the imposition of a larger initial subsidy has changed the “social norm” – which can then
replace the subsidy as the primary force that leads people to drive green cars.
(g) Sometimes people advocate for so-called “sin taxes” – taxes on such goods as cigarettes or pornography.
Explain what you would have to assume for such taxes to be justified on efficiency grounds in the absence
of network externalities.
(h) How could sin taxes like this be justified as means of maintaining social taboos and norms through network
externalities?
B: Suppose you live in a city of 1.5 million potential car owners. The demand curves for green cars x for
“greenies” and “meanies” in the city are given by xg (p) = (D − p)/δ and xm (p) = (A + BN 1/2 − p)/α, where
N is the number of green cars on the road and p is the price of a green car. Suppose throughout this exercise
that A = 5, 000, B = 100, D = 100, 000, α = 0.1 and δ = 5.
(a) Let the car industry be perfectly competitive, with price for cars set to marginal cost. Suppose the
marginal cost of a green car x is $25,000. How many cars are bought by “greenies”?
(b) Explain how it is possible that no green cars are bought by “meanies”?
(c) Suppose that the purchase of a green car entails a positive externality worth $2,500. For the case described
in (a), what is the impact of a Pigouvian subsidy that internalizes this externality? Do you think it is
likely that this subsidy will attract any of the “meanie” market?
(d) Would your answer change if the subsidy were raised to $5,000 per green car? What if it were raised to
$7,500 per green car?
(e) ** Suppose that a subsidy of $7,500 per green car is implemented and suppose that the market adjusts to
this in stages as follows: First, “greenies” adjust their behavior in period 0. Then, in period 1 “meanies”
purchase green cars based on their observation of the number of green cars on the road in period 0. From
then on, each period n, “meanies” adjust their demand based on their observation in period (n − 1).
Create a table that shows the number of green cars xg bought by “greenies” and the number xm bought
by “meanies” in each period from period 1 through 20.
21B. The Mathematics of Externalities
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(f) Explain what you see in your tables in the context of network externalities and changing social norms.
(g)
* Now consider the same problem from a slightly different angle. Suppose that the number of green cars
driven by “greenies” is x. Then the total number of green cars on the road is N = x + xm . Use this
to derive the equation p(xm ) of the demand curve for green cars by “meanies” – and illustrate its shape
assuming x = 16, 000.
(h) Relate this to the notion of “stable” and “unstable” equilibria introduced in exercise 21.8B(e). Given
that you can calculate x for different prices, what are the stable equilibria when p = 25, 000? What if
p = 22, 500 and when p = 17, 500.
(i) Explain now why the $2,500 and $5,000 subsides would be expected to cause no change in behavior by
“meanies” while a $7,500? would cause a dramatic change.
(j) Compare your prediction for xm when the subsidy is $7,500 to the evolution of xm in your table from
part (e). Once we have converged to the new equilibrium, what would you predict will happen to xm if
the subsidy is reduced to $2,500? What if it is eliminated entirely?
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Chapter 21. Externalities in Competitive Markets